r 


ev5 


LIBRARY 


UNIVERSITY   OF   CALIFORNIA. 

GIKT  OK 

Received  K^^^a^^S^^iSS^ 

Accessions  No.  ^<^^  §"/  .S"-^^//  iV^. 


fib 


ROBINSON'S    MATHEMATICAL    SERIES, 


KEY  TO   ROBINSON'S 


NEW 


UNIVERSITY   ALGEBRA. 


FOR  TEACHERS   AND   PRIVATE   LEARNERS. 


NEW    Y  O  K  K  : 
IVISON,    PHINNEY,    BLAKEMAN    &  CO., 

CHICAGO:    S.    C.    GRIGGS   &    CO. 

180  7. 


ROB  INSON'S 

27i€  most  COMPLBTB,  most  Practical,  and  most  Scientific  Series  of 
Mathematical  Text-Books  ever  issued  in  this  country 

{IN  TWKNTY-X^VO  VOLTJMiEJS.) 


1  Robinson's  Proan^essive  Table  Book,  • 

II.  Bobinson's  Progressive  Primary  Arithmetic,- 

III.  Robinson's  Progressive  Intellectual  Arithmetic, 

IV.  Robinson's  Rudiments  of  Written  Arithmetic, 
V.  Robinson's  Progressive  Practical  Arithmetic, 

VI.  Robinson's  Key  to  Practical  Arithmetic,  - 

VII.  Robinson's  Progressive  Higher  Arithmetic,    • 

VIIL  Robinson's  Key  to  Higher  Arithmetic,     - 

IX.  Robinson's  New  Elementary  Algebra, 

X.  Robinson's  Key  to  Elementary  Algebra,  - 

XI.  Robinson's  University  Algebra,   .       -       .       . 

XIL  Robinson's  Key  to  University  Algebra,    - 

XIII.  Robinson's  New   University  Algebra, 

XIV.  Robinson's  Key  to  New  University  Algebra,  • 
XV.  Robinson's  New  Geometry  and  Trigonometry, 

XVI.  Robinson's  Surveying  and  Navigation,     - 

XVII.  Robinson's  Analyt.  Geometry  and  Conic  Sections 

XVIII.  Robinson's  Differon.  and  Int.  Calculus,  (in  preparation,) 

XIX.  Robinson's  Elementary  Astronomy,   - 

XX.  Robinson's  University  Astronomy,     • 

XXI.  Robinson's  Mathematical  Operations, 

XXII.  Robinson's   Key  to   Geometry  and  Trigonometry, 
Sections  and  Analytical  Geometry, 


Conia 


Entered,  according  to  Act  ot  Congress,  in  the  year  1862,  by 

DAKIEL  W.  FISH,  A.  M., 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Korthern 
District  of  New  York. 


PREFACE. 


This  volume  contains  solutions  of  nearly  all  the  examples 
found  in  Robinson's  New  University  Algebra.  The  greater 
part  of  these  solutions  might  have  been  omitted,  if  the  sole 
object  of  a  Key  were  to  aid  teachers,  of  limited  acquirements 
and  experience,  in  overcoming  difficulties  otherwise  insur- 
mountable by  them.  Though  this  may  be  one  of  the  primary 
objects  of  a  Key,  there  is,  nevertheless,  a  higher  and  far  more 
important  use. 

This  higher  object  is  purely  educational,  and  is  simply  an 
extension  of  the  author's  design  in  the  solutions  given  in  the 
text-book.  It  is  to  illustrate  more  completely  the  principles 
of  the  science,  in  their  practical  bearings  ;  and  to  show  the 
application  of  the  rules  and  methods  taught,  under  the 
greatest  possible  variety  of  circumstances.  Viewed  in  this 
light,  and  used  with  discretion,  a  Key  to  an  Algebra  may  be 
serviceable  to  all  students  of  the  science,  and  especially  to 
teachers  who,  by  reason  of  limited  educational  advantages, 
find  it  necessary  to  apply  themselves  diligently  to  self-culture 
in  their  profession. 

These  remarks  apply  more  particularly  to  those  examples 
which  are  designed  to  tax  the  ingenuity,  and  to  call  forth 
algebraic  skill.     In  the  case  of  examples  requiring  long  and 


iv  PREFACE. 

tedious  numerical  calculations,  a  Key  is  often  a  great  con- 
venience in  detecting  mistakes,  the  correction  of  which  would 
require  from  the  teacher,  under  the  pressure  of  school-room 
duties,  a  needless  expenditure  of  time  and  labor. 

We  have  given  full  solutions  of  the  numerical  equations 
of  higher  degrees,  found  on  the  last  page  of  the  Algebra, — a 
feature  of  the  work  which  will  be  acceptable  to  all  who  use 
the  book.  The  attention  of  teachers  is  invited  particularly 
to  the  system  of  decimal  contraction  applied  in  these  solu- 
tions, and  to  the  convenient  method  of  marking  the  cor- 
responding terms  in  the  several  columns. 

September,  1862. 


■:■■  ■;^ 
kSyto 


ROBINSON'S 


NEW  UNIVEKSITY  ALGEBRA. 


USE  OF  THE  PARENTHESIS. 
(61,  page  29.) 

1.  3a  +  {2b^—a—d  +  m)  =  da  +  2b^—a—d-\-m 

=  2a4-2i' — d-\-m^  Ans. 

2.  4a;'— y— (3a:— 7y  +  5)4-2ar=:4j;'— y— 3a:  +  7y— 5+2a: 

=  4a;'4-6y— a;— 5,  ^W5. 

3.  a  +  2c  —  (4c  —  3a  +  2m') =a  +  2c— 4c  +  3a  — 2m' 

=  4a  — 2c  — 2m',  ^n5. 

4.  4i:'— 2a:'  —  [x»— (2a:' +  5a;— 7)— 6a:  +  l]  =  4a:«—  2a:'— a:' + 
(2a:' +  5a;— 7)  +Qx—\  =  4a:'— 2a;'— a;'+ 2a:' +  5a;  —  7  +  6a;  —  1  = 

3x*  +  \lx  —  Q^  Ans, 

5.  a  +  2m— Jc+a;- [a— m— (c  — 2a:)]j 
=  a4-2m  — c— a;  +  [a— m— (c  — 2a;)] 

=  a4-2m— c— aj  +  a— m— c4-2a;  =  2a  +  m  — 2c+a',  Ans. 

6.  3a;'  — 4.1;— am—  j.r'— a;  — [3am  — (2a; 4  2am)  +  2a;']  — 5am j 
=  3a:'  — 4a;— «m— a:'  +  a:  +  [3am  — (2a:  +  2am)  +  2i:']  +  5am 
=  3a;'— 4a;— am— x'  +  a;  +  3am  — 2a;— 2om  +  2a:'  +  5am 

=  4a;'  — 5a:  +  5am,  Ans. 

7.  3a— J2m»  +  [5c— 9a— (3a+m')]  +  6a  — (m'  +  5c)J 
=  3a  — 2m'  — [5c  — 9a  — (3a  +  m')]  — 6a  +  m'  +  5c. 

=  3a— 2m'  — 5c  +  9a  +  3a-4-m'  — 6a  +  m'  +  5c  =  9a,  Ans. 
(29) 


FACTORING. 

8.  a;'— S5mc'— [j:'— (3c— 3r.ic')  +  3c  — (j-'  — 2mc'— c)]j 
=  x^  —  5mc^-h[.c'  —  (3c-3mc')-^Sc  —  (.v'  —  2mc^—c)] 
=  a;'  — 07wc'  + j;'  — 3c  +  3;?ic''  +  3c— x'  +  2mc'  +  c 

=  x'  +  c,  ^n*. 

9.  m'  —  m  —  l—\ m''  —  2m  — 2  —  [«i'  —  3;?i  —  3  —  (m'  —  4;/i  —  4)] \ 
=  m^—m  —  l—m^  +  2m  +  2  +  [7?i''  — 3m  — 3  — (m''  — 4m  — 4)] 
=  m' — m  —  1  —  7n'  +  2 m  +  2  +  m'  —  3  ;n  —  3  -  m'  +  4  ??i  +  4 

=  2rn4-2,  Ans, 

10.  52»_3z'  +  42-l-[22'-(32'  — 22:  +  l)-0'  +  2] 
—  5z'  —  3z^  +  42—1  —  22'  +  32'  — 22  +  1+2'  — z 

=:  42* +2,  ^n*. 

11.  4c'-2c'  +  c  +  l-(3c'-c'-c-T)-(c'-4c'  +  2c  +  8) 
=  4c'-2c'  +  c  +  l  — 3c'  +  c'  +  c  +  Y-c'  +  4c'-2c-8 


12.  3a'6  — 4ccf— (3c</— 2a'6)  — [a»  +  c— (5crf  +  3a'6)-f  (3a'  +  2crf) 
4-a']  z=z  Sa'b  —  4cd  —  3cd+  2a'b—a^—c  +  (5cd  -f  3a'6)  — (3a'+  2cd) 
— a'  =  3a'6— 4cJ— 3rflr  +  2a'6  — a'  — c  +  5cc?  +  3a'6  — 3a'  — 2crf— a* 

—  8a'6  — 4c(/— 5a'— c,  Ans, 

13.  —  (4a'77i  +  3mV— (7w'c?  — 9a'm— n)— J5n— [wV— (2n  + 
a'm)  +  San']  -  5a'm  |  —  1  2a'm)  =  _  4a'm  -  37?iW  +  (7m'cf  — 
9a'm  —  n)  +  {on  —  [in^d—  {2n  +  a'w)  +  3aw']—  5a'mf  +  12a'm 
r=:  —4,a^m~  Sm'^d  +  T/nV  —  9a^m  —  7i+5n  —  [m''d—(2n  +  a'm)  + 
3a'n]  — 5a'w  +  12«'m  =  — 4a';w—  3m'(/+  Im'd—  9a^m—n  +  5n^ 
m*d  +  2»  +  a'w  —  3an'  —  5a'w  + 1 2a'wi 

=  3wV  +  6n— 5a'm  — 3an',  ^w*. 


FACTORING. 
(95,  page  51.) 

5.  x*—x^y-\-xy*—y*  =  z\x—7/)  +  7/\x—7/)  =  {x^-}-i/)(£—i/\  Ans, 

6.  a*b'  +  2a'b'-{-a'b'  =  a'b\a^ -]-2ab  +  b') 

=  a''b\ai-b)(ai-b),An8, 
(  29-51 ) 


SUBSTITUTION. 


7.  {x*—x)a+  {z*-{-x)(db—c)  —  q  =  ax"—  ax  +  36j;'+  Zhx—cx*— 
cx—q  =  ax*-{-3bx*—cx^—ax  +  3hx  —  cx  —  q 

—  (a  +  3b  —  c)x^  —  (a  —  Sb  +  c)x—q,  Ans, 

8.  a*m— 9am'=a7n(a*  — 9w')  =  a7?i(a'-f  3m)(a*— 3m),  Ans. 

9.  By  (89,   4),  8a*-x*  =  {2a-x){-iu^  +  2ax-\-x'),  Ans. 

10.  y'  +  243   is  the  sum  of  the  fifth  powers  of  y,  and   3  ;  hence 

by  (89,  1), 

/  +  243=:(y  +  3)(/-3y«  +  9y'-2'7y  +  8l),  ^715. 

11.  By  (70,   III.),  we  have 

z'-y-  =  (i:«  +  y')(.«:'-/). 
And  by  (89,  1  and  4), 
(x'-}-y')(x'-i/*)=.{x  +  y){x'-xi/  +  y'){x~-y){x'  +  xi/+^'),Ans, 

12.  a'  —  ab^  +  2abc—a^  =  a(a^  —  b'  +  2bc—c') 

=:a\a^-{b-cy\ 
Or  by  (70,  III.),  =  a(a  +  6-c)(a— 6  +  c),  Ans. 


SUBSTITUTION. 
(96,   page  51.) 

a*=(a-by  =a'-2ab  +  b' 
ab={a-b)b=  ab-b' 

b*  =  6' 


a'  +  ab  +  b^  =a^—ab-[-b%  Ans. 

2.  a*=z     (.r  +  2)' =x'  +  4a;  +  4 
—  2a=— 2(.c+2)=     — 2ar— 4 

1 = 1 

a"  —  2a  +  1  =a:'  +  2a;  + 1,  Ans. 

3.  y*  =  {x  +  Sy       =a;*  +  12x*  +  54a:'  +  108^  +  81 
—  2y*=  — 2(a:  +  3)=         -2a:*-18r'— 54a;-54 

f  =  {x  +  sy       =  x'  +  6x  +  d 

-6 = -6 

y*_2y«  +  y»  — 6    =a;*  +  10i;'  +  37j;''  +  60.c  +  30,  vlw*. 
(61) 


6 


SUBSTITUTION. 

x«=(»+r)'  =«'  +  2sr+r» 
ax=(s-{-r)a=  as-\-ar 
b  =  h 


or  =a 

a*b=a*  xa  =d 
a^b*=a^  xd*=a 
ab*=a  X  a'=a 
b*  =a 


a*  +  a'b  +  a*b^  +  ab'  +  b*z=  5a\  Ans, 

ax^=(Tn  —  l){m-\-iy  =  m'+   m'—  m—l 

a'x=(7n  —  iy(m-\-l)=m'—  iii^—  m  +  \ 

a'=(m  — 1)'  =m*  — 3w'  +  3w  — 1 

«*+/  =:2a*  +12??  +26* 

=  2(a*  +  6a'6'+6*),  ^n*. 

8.  When   a  +  i+c=5,  ar  +  a  +  6  +  c=ar+5,  and  a;— a  — 6^c=a;  — 
(a  +  6  +  c)-=a;— 5. 

(a;+a  +  6+c)»=(a;  +  s)»=:a;*  +  5^*s  +  10xV  +  10xV  +  5ar«*+5» 
(ar— a  — 6  — c)'=:(a;— 5)'=x'  — 5ar*5+10xV— 10xV  +  5a-s*— «• 

(jr  +  a+6+c)'  +  (ar— a  — 6— c)'=2^'  +20a:V  +10a:«* 

=2(a;'  +  10a?V  +  5a-«*),  ^n5. 


x*=        (y_2)''=y'-6y'  +  12y-8 

—  7ar=-7(y-2)  =  — 7y  +  14 

6  =  6 

ar* — 7a:  +  6  r=:y'  —  6y'  +  5y,  ^n5. 

(52) 


GREATEST    CO 

10.  x'=       (y+l)*=/  +  5/  +  1^5yTTOF  +  5y  +  l 
_2.c*  =  — 2(y  +  l)*=       — 2/  — 8/— 12y'  — 8y  +  2 

3.c»=      3(i/  +  iy=  3/4-9/  +  9y  +  3 

8x=     8(y  +  l)=  8y  +  8 

-3  = -3 

x*—2x*-^dx*  —  1x^  +  8x—3  —  7/''  +  Si/*-{-5f/%  Ans. 

11.  Expand  before  substituting,  and  wo  have 
2{a—by{b-cy  =  2a'b'-4.ab'  +  2b*-Wbc  +  8ab'c-4b'c  +  2a'c'- 

[4abc'-\-2b^c* 
2{a-bY(c—ay  =  2a'c^  —  4abc'-^2b\*^4a'c+8i''bc  —  4ab'c  +  2a*  — 

[4a'h  +  2a^b^ 
2{b—cy{c—ay=:2b'c'  —  4bc'  +  2c*  —  4ab'c  +  8a0c'  —  4ac'  +  2a''b'  — 

[4a'6r  +  2aV 
If  we  add  the  quantities  thus  obtained,  all  the  terms  containing 
three  letters  each  will  disappear.  Arranging  the  terms  containing  a 
and  b  according  to  their  powers ;  also  the  terms  containing  b  and  c 
according  to  their  powers ;  also  the  terms  containing  a  and  c  ac- 
cording to  their  powers, — observing  to  separate  the  terras  2a*,  26*, 
and  2c*,  into  parts,  we  have  three  polynomials,  as  follows : 

a*-4a'b  +  Qa'b'-4ab'-\-b*=  {a-by=zx* ; 
h*-4b*c  +  Qb'c*  -4^c'  +  c*=  (6-c)*=/; 

c*-4c'a  +  6cV  — 4ca'+tt*=(c-«)*=2;*j 

Sum  =:a;*-j-y*+2*,  Ans, 


GREATEST   COMMON   DIVISOR. 
(lOO,   page  53.) 

2a' be' =        2a'bc' 

6ab'c'  =  Sx2ab'c'' 

10a'6c'  =  5x2a'6c' 


Hence, 


Hence, 


2a6c',  Ans, 


6x'i/'z*=5xYz' 
Gx'i/z*  =3  X2ar»y2' 
I2x^i/z^  =2x2  xSx^'ijz' 

(52-63) 


x^i/z^y  Ana, 


10 


4. 


GREATEST   COMMON   DIVISOR. 
x^-,/  =  {x  +  y){.t-y) 


x*-2xy  +  y^  = 


Hence, 


{^-y){^-y) 


lie  nee, 


Hence, 


V. 


Hence, 


x—y^  Ans, 

a'  m  —  h^m  =    m((i  —  h)  (a + b) 
2'ic^m  —  2bc^7nz=z2c^  X  7n(a  —  b) 

m(a— 6),  Ans. 

aV  — 3aV  +  aV=      a^x{x^  —  3x+\) 
daxz^  —  ax'^z* —az'=—  az'lx'  —  3x  + 1 ) 

a(x'  — 3a;Xl),  Arts, 

lCa;'-l  =  (4i:4-l){4a:-l)  (by  70,  III.) 

l-8a;+16x'=  (4a^— l)(4x-l)      "       "      II. 

4^—1,  Ans, 

(105,  page  60.) 

FIRST    OPERATION. 

a:*  — 2x'—   4i:'+    lU-   eix'-S-r'  +  lYar-lO 


a:*_8x'H-17j:' 


10.r 


.r  +  6 


G.r'  — 21^'+    2U—   6 

Or'  — 48j;'  +  102j-  — 60 

27x'~   81a:  +  54,  1st  remainder. 
Dividing   iliis  remainder  by  27,  we  have  a;'  — 3j;4-2  for  the  next 
divisor. 

SECOND    OPERATION. 


a:'  — 8j:'  +  17j;— 10 
.r'  — 3jr'+    2x 


j:'_3ar4.2 


a;— 5 


—  5a;'  +  15ar— 10 

—  5.r*  +  15.r— 10 

'WTience,  a:'  — 3a; +2,  Ans. 

2.  No  pteparation  is  necessary  for  the  first  operation,  and  as  a;  is 
involved  to  the  same  power  in  both  of  the  polynominals,  it  is  imma- 
terial which  is  taken  as  a  divisor. 

1st. 
6x'+     a:'  — 44ar  +  21l6a-'  — 26a:'+46a:— 42 


6a;'  — 26a;' +  46a;— 42 1 1 
27a;'  — 902;  + 63 

(53-60) 


GREATEST    COMMON    DIVISOR.  11 

Dividing  this  remainder  by  9,  we  have  dx*  —  10x-\-^  for  the  next 
divisor. 

2d. 


6a;'— 26j;'  +  46a;-42 
6a;'— 20a:' +  14a; 


3a:'-10a;  +  V 


2a-— 2 


—  6a;'  +  32a;-42 

—  6a;' 4- 20a;— 14 

12a;-28 
Dividing  this  last   remainder  by  4,  wo  have  3a;— 7  for  the  next 
divisor. 

3d. 


3a;'— 10a;  +  7 

3a;- 7 

3a;'-   1x 

a;-l 

-  3.C  +  7 

—  3a;  +  Va; 

Hence,  3a:— 7,  Ans» 

3.  The  greater  polynomial  must  be  multiplied  by  3a,  to  render  its 
first  term  divisible  by  the  first  term  of  the  less  polynomial? 

1st. 
3aa;'  —  ISa'a;'  +  30a'a;  —  9a*[i3aa;'  —  14a'a;  +  1 5a' 


3ax'  — 14a'a;'  +  15a'a;  [     r,      —4a 

—  4aV  +  15a''a:-9a* 

—  12a'a;'  +  45a'a;  — 27a*,     new  prepared  dividend. 

—  12a'x'  +  56a*a;-60«* 

—  lla'a;  +  33tt* 

a;— 3a,     next  divisor. 

In  the  above  operation  the  first  remainder  is  multiplied  by  3,  to 
render  its  first  term  divisible  by  the  first  term  of  the  divisor.  The 
last  remainder  is  divided  by  — 11a'  for  the  next  divisor. 

ad. 


3aa;'  — 14a'a;  +  15tt' 
3aa;'—   9a'a; 


ar— 3a 


3aa; — 5a' 


—  6a'a;+15a' 

—  5a'a;+15a' 

Hence,  a;— 3a,  Ans. 
(60) 


12 


GREATEST   COMMON   DIVISOR. 


1st 

x*—8x'  +  l9x^  —  i4z         \x 
—    5x'  +  30.r  — 40 


Mviding  by  —5,  we 

have  X*- 

-62*4-8  for  the  second  divisor. 

2d. 

3d. 

a:'-8x'4-19j;  — 14 

x'- 

X- 

-6^4 

-2 

-8 

a;«_6x4-8 
x^-2x 

-X4-2 

'a;'-6.c'  +  8.r 

-X4-4 

-2x''4-ll-r-14 

—  4x4-8 

-2:t'  +  12jr-16 

-4x4-8 

-x  +  2 

Hence, 

— X4-2,  or  X— 2,  Ans, 

1st. 

ad. 

a'4-5a'4-5a4-l 
a'  4-1 


a'4-1 


a"  4-1 
a*-\-a' 


a  4-1 


a'  — a4-l 


5a' 4- 5a 
Suppress  (5a),  and  we  have  a 4-1 
for  the  next  divisor. 


6.  Multiply  the  first  polynomial  by  2. 

1st. 
4a*-- 10a*6- 6a'6' 4- 14a6*  4- 66* 
4a* -^  2a'6-4a'6'-    Sab' 


-a' 4-1 


a  +  1 

a  +  l 

Hence,  a  4- 1,  Ans, 


4a*-2a'b-4ab'-3b* 


a  —2b 


—  8a"6  -  2a'6''  +  1  lab'  +  6b* 

—  8a'64-4a'6'4-    8a6'4-6ft* 


Suppress  —  3a6% 
and  we  have, 


—  6a'6''4-9a6' 

2a  — 36     for  the  next  divisor. 


ad. 

4a'  — 2a'6  — 4a6«  — 36»j2a— 36 
4a' 


6a'6 


2a"4-2a64-6* 


4a'6— 4a6' 

4a'6-6a6' 

2^ 

2a6'. 


36' 
36' 


Hence,  2a— 36,  Ans, 


(60) 


GREATEST   COMMON   DIVISOB,  13 

7.  Multiply  the  greater  polynomial  by  4,  to  render  its  first  term 
divisible  by  the  first  term  of  the  other  polynomial. 


1st. 
1 2^'  —  1 6  j:  V  +  1 2.ry'  -  8y' 
12x'  — 21arV+    9.r/ 


4.r'  — 7ary  +  3y' 


Multiply  by  4.       bx'y  +    ^xy^—   8y' 

20j:'y  +  12a:y'  — 32y'     New  prepared  dividend. 
20a;V— 35a;y'  +  15y' 
47x/  — 47^' 
Dividing  by  47y',  we  have  a:— y  for  a  new  divisor. 

3d. 
4a;'  — 7ary  +  3/ 
4a;'  — 4.ry 


4a:-3y 


—  3a:y  +  3y' 
— 3a;y  +  3y' 


4ar*—  2a;*  +   4a;'— 27^'+    4a;— 7 
4a;'H-12a:*  — 38.t'+    8a;'  — 10a; 


Hence,  x—y^  Ans, 

2ir*  +  6a;'  — 19a;'  +  4a;— 5 


2a;- 7 


—  14x*4-42a;'—   35a;' +  14a:- 7 

—  14j*  — 42a;'4-133a;'  — 28a;+35 

84a;'^— 168a;'  +  42a;— 42. 

Dividing  the  remainder  by  42,  we  have  2a;'— 4a;'+a;— 1  for  the 
next  divisor. 

ad. 
2a;*  +  6a;'  — 19a;'  +  4a;— 5|2a;'-4a;'+a;— 1 
2.r*— 4a;*-t-      x*—x         |    a;+5 
10a;'  — 20a;' +  6a;  — 5 
10a;'  — 20a;' +  5a;— 5 

Hence,  2a;'— 4a;'+a;— 1,  Ans. 

9.  The  first  polynomial  contains  the  monomial  factor  ac^  and  the 
second  contains  the  monomial  factor  c'.  Suppress  these  factors,  and 
set  aside  €,  which  is  common  to  both,  as  one  factor  of  the  greatest 
common  divisor;  then  apply  the  process  of  division  to  the  resulting 
polynomials. 

(60) 


14 


GREATEST   COMMON   DIVISOR. 


1st. 


a*  —  ea*m-h5m* 


Dividing  tho  remainder  by  2wi,  we   have   a*—m  for  the  next 


divisor. 


ad. 

6a'm  +  5m'  «'  —  m 


a 

o.'—  a'm 


6m 


— 5a'm4-6m' 
— 5a'm4-5m' 

Hence,  c(a* — m),  Ans, 

10.  Multiply  the  greater  polynomial  by  2,  to  render  its  first  term 
divisible  by  the  first  term  of  the  other  polynomial. 

1st. 
2x*-'   8x'—   32a;' +    14j:-f    48  2^'  — 15ar'4-0ar  +  40 
2x*— 15x'+      9j;'4-   40.C 


X,  +7 


7x'—  41.C'—  26x+  48 
14x»—  82a:'—  52x+  96 
14j:'  — 105x'4-    63j'  +  280 


Kew  prepared  dividend. 


23j;'  — 115a;— 184 
Divide  by  23,  and  we  have  a;'  — 5a;— 8  for  the  next  divisor. 

«d. 


2a;'  — 15r'+   9.i;  +  40 
2a;'  — lOa;'  — 16a; 


5a;— 8 


2a;— 5 


11. 


—  5a;' +  2  5a;  4- 40 

— 5a;'  +  25a;  +  40       Hence,  a;'  — 5a;— 8,  ^n*. 

1st 

15a;*  +  7la;*+    60a;'—   66i3a;"  — l7a;*-20a;'  +  84 
15a;"  — 85a;*  — 100a;'  +  420 


Suppress  4,  156a;*  +  160a;"  — 476 

and  we  have  39a;*  +   40a;'  — 119     for  the  next  divisor. 

Multiply  the  first  divisor  by  13  to  render  division  possible,  and 
proceed  as  follows : 

(60) 


GREATEST    COMMON    DIVISOR. 


15 


ad. 

Z9x'  —  22lz*~-   200ar'  +  1092 
39a:''+   40x*—    119^' 


39a?*  +  40ar'-119 

x\  +29 


Suppress  —3,  —261a;*—    14  It' +1092 

and  we  have  Slx*-{-     47a;'—   364 

Multiplymgbyl3,wehave  1131a;*  +    61  la;'  — 4732      New  prepared  dividend. 
1131a;* +  1160.g'  — 3451 
-549.t'  — 1281 
Suppress  the  factor  —183,  and  we  have  3x'  +  7  for  the  next  divisor. 


3d. 

39a;*  +  40a;'  — 119 
39a;* +  9  la;' 


3a;' +  7 


I3a;'-17 


—  51a;'— 119 

—  51a;'-119 

Whence,  3a;' +  7,  Ans. 

12.  We  will  first  find  the  greatest  common  divisor  of  the  first 
two  polynomials,  using  the  second  as  dividend,  and  the  first  as 
divisor. 


1st. 

6a*  — 14a'm'+   4m\ 
6a*  +  28o'm'  — 10m* 


3rt*  +  14a'»i'--6m* 


—  42a'm'  +  14m* 
Suppressing— 14m',  we  have  3a'  — m'  for  the  next  divisor. 


2d. 

3a*  +  14a'm'  — 57?^*■3a•• 
3a*—     a^rn* 


m' 


I  a'  +  5m' 


+  15a'm'  — 5m*- 
ISa'm'  — 5m* 

Hence,  3a'— m'  is  the  greatest  common  divisor  of  the  first  two  poly- 
nomials. We  must  now  find  the  greatest  common  divisor  of  this 
result  and  the  third  polynomial,  which  will  be  the  greatest  common 
divisor  required. 

(60) 


16  GREATEST   COMMON   DIVISOR. 


3a*—     a^m* 


3a' 


7m» 


-21a'/w'  +  Vm* 

Whence,  3a'  — m',  ^na. 

13.  The  second  polynominal  contains  the  niononiinal  factor  bi/. 
Suppressing  this  factor,  we  use  the  resulting  polynominal  as  the  first 
divisor. 

1st. 


2aV— 2tt'teV+   oh'xy*-   6'/ 
2a  V — Aa^hx'y  +  2a6'a-y' 


a^x*  —  2ahxi/-{-b^y* 


2ffa?-f  2Ay 


2aV,x^y  —  4ah\ry^  +  2bY 
3<'b''xy^-3h'y' 
Suppressing  36'y'  we  have  ax— by  for  the  next  divisor. 

2d. 
a^z* — 2abxy  +  h^y^ax — by 


a'x' —  abxy  lax  —  by 

—  abxy  +  b^y* 

—  abxy  +  b^y' 

Whence,  ax— by,     Ans, 

1st. 
14.  9a*  4- 12a' +  10a' +    4a  +  l;3a*  +  8a»  +  14a'4-8a4-3 


9a*  +  24a'  +  42a'  -f  24a  +  9 ,3 


Suppressing —4,      —12a'  — 32a'  — 20a  — 8 

we  have  3a'  +    8a'  +    5a  +  2  for  the  next  divisor. 

ad. 
3a*  +  8a' +  14a' +  8a+3|3a' +  8a' +  5a  +  2 


3a*  +  8a'+    5a'  +  2a 


Suppressing  3,  9a'  +  6a  +  3 

and  we  have  3a'  +  2a  + 1  for  the  next  divisor. 

(60) 


Whence, 


LEAST   COMMON   MULTIPLE. 

3d. 

3a'  +  8a'  +  5a  +  2i3a'  +  2a  +  l 
3a'  +  2tt'+   a        a  +  2 


17 


6a' +  4a +  2 
6a'4-4a  +  2 


3a'4-2a  +  l,  Ans, 


2. 


Hence, 


3. 


Hence, 


Hence, 


LEAST   COMMON  MULTIPLE. 

(109,   Page  62) 
2a*hc  —  2a'bc 

I0ab'd=2x5ub'd 
l5abcd  =  S  x  5abcd 


2  X  3  X  5  X  a*6Vc?=30a*6V(/,  Ans. 

32r'y=3x'y 
15xf/^  =  3  X  5  xry' 
lOxyz' z^ 2  X  5  X  zyz* 
5x'y^zz=z5x'i/^z 

2  X  3  X  5xYz^=^0xYz\  Ans, 

x*  +  xr/=x{x  +  j/) 

{x  +  y)(x—y)xy—x^y—xy\Ans. 

x^—a*  =  {x'  +  a'')(x-\-a)(x—a) 
x'*—a^=  (x-^-a)  [x—a) 

x'  +  a'  =  {x'  +  a') 
2a'ar'  +  rt*=  {x  +  aY{x-ay 


Hence, 


(x'  +  «')  (x-\-ay{x-ay=x'-a'x*--a*x*  +  a*,  ^n«. 

6.  x*-x=      a:(.r  +  l)(:c-l) 

:i:'-l=(^»+ar+l)(.r-l) 
ar'4.1=(a;'-ar4-l)(ar+l) 

Hence,    x{x'+x  +  l)(x'-x+l)(x-^l)(x-l)=zx(x'—l)=x'^x, 

Ans. 
(62) 


18  LEAST   COMMON   MULTIPLE. 

X^  +  2X+1=:  (^+1)' 

a:'  — 2jr  +  l=  (^-1)' 

x  +  l=  {x+1) 

ar-l=  (jr— 1) 

Hence,  (x'  +  iy{x  -f  \y{x-iy=x'-2x*  + 1,  ^»5. 

8.  4x»  +  2a'=2.r(2x'  +  l) 
6j:'-4jr=2r(3a:-2) 
6j:'  +  4jr=:2.r(32:  +2) 

Hence,    2x{2x'-{-\)(3x  —  2)(3x-\-2)  =  36x^  +  2x'  —  8x,  Ana, 

9.  ar'  — 4a'  =  (.c  +  2ff)(j:  — 2«) 
(ar  +  2a)'  =  (x+2a)' 
(x-2ay=  {x-2ay 

Hence,  (j:  +  2a)'(j:— 2a)'=(i:'  — 4a')*=x'— 12j:V-+-48xV  — 64a', 

10.  a*-b*=^{a'  +  P)(a  +  b)(a-b) 

a'-b'=  la-b){a'  +  ab  +  b') 

a'-b'z=  (a  +  i)(a-6) 

a-b=  (a-b) 

Hence,  (a'  +  6')(a  +  6)(a-6Xa'  +  a6  +  6')=o'+a'6+a*6'— a'6*-a6''— 6', 

Ans. 

(110,  page  63.) 

1.  The  greatest  common  divisor  of  the  two  polynomials,  (  10«5), 
is  a:'  — 2.C  +  2  ; 

(x*  —  5x*  +  8x—G)-i-(x'  —  2x  +  2)=x—3  ;  hence, 
lx-3){x'+x'  —  ix  +  6)=x'-^2z*  —  W  +  18x—l8,  Ans. 

2.  The  greatest  common  divisor  of  the  given  polynomials,  (105), 
is  ar  — 5 ; 

(x'-l2x-{-35)^(x-5)=x-1;  hence, 
(x-1){x'  —  2x^  —  }dx  +  20)=x*  —  9x'  —  5x^  +  l53x—l40,  Ans. 

3.  The  greatest  common  divisor  of  the  given  polynomials,  (I05), 
is  2am'  —  1 ; 

(63) 


LEAST   COMMON   MULTIPLE.  19 

(2a'wi*  +  3aw'  — 2)-7-(2ow'— l)=am*  +  2  ;  hence, 

{am^  +  2){6a^m*—am''  —  l)  =  Qa'm^-\-Ua'm*  —  Sam*  —  2,  Ans, 

4.  The   greatest    common    divisor  of   the   given  polynomials  is 

{x'  —  5x*  +  1x  —  2)^{x'  —  3x-hl)=x  —  2;  hence, 
{x-2){2x'-5x'—x-hl)z^2x*-9x'  +  9x'i-3x-2jAns. 

5.  Tlie   greatest   common   divisor  of    the   given    polynomials   ie 
3j:'-5; 

(3x'  +  ex'  —  5x— 10)-^  (3^'  —  5)  =a:  +  2  ;  hence, 

lx  +  2){6x*  —  4x'-l0)  =  Qx*  +  12x*  —  4x*  —  8x'  —  l0x—20,  Ans. 

6.  The  greatest  common  divisor  of  first  two  polynomials  is  a: 4- 2 ; 

(^x'-2x-8)-^(x  +  2)  =  (x-4)', 
hence,  {z—4)(x*  +  7x  +  lO)=zx*-\-Sx*  —  l8x-40,  the   least  common 
multiple  of  the  first  two  polynomials.     We  now  proceed  to  find  the 
least  common  multiple  of  this  result,  and  the  third  polynomial. 
The  greatest  common  divisor,  <fec.,  is  a;'-f  ar— 20; 
(.r'-fa;— 20)-^(J:'^-x  — 20)  =  1  ;  hence, 

{x*  +  dx*  —  l8x—40)x\=x*  +  3x*  —  18x—40y  Ans, 

*l.  Tlie  greatest  common  divisor  of  the  first  two  polynomials  is 
a— 26;  and 

{a}  —  ah  —  2h'')-^{a  —  2h)=a-\-h\  hence, 

{a-[-h){a*-3ab  +  2h')  =  a'-2a'b-ab\+2h\ 
which  is  the  least  common  multiple  of  the  first  two  polynomials. 

The  greatest  common   divisor  of  this  result  and  the  third  poly- 
nomial is  a'  — 6';  hence, 

a»_2a'6— a6'  +  26',  Ans. 

8.  The  greatest  common  divisor  of  the  first  two  polynomials  is 
2.T— y,  and  (22:'— 5xy-|-2y')-7-{2.r--y)=a;— 2y.     Hence,  we  have 

(x-2y){2x''-1xyJc  3y')=  2x*—  lla;'y+  I7.ry'—  6y', 
which  is  the  least  common  multiple  of  the  first  two  polynomials. 

The  greatest  common  divisor  of  this  result  and  the  third  poly- 
nomial is  x^—bxy  +  Qy'* ;  hence, 

2x*—\\x'y-ir\1xy^  -6y\  Ans, 
(63) 


20  FRACTIONS, 

FRACTIONS. 

REDUCTION. 

(1245  page  67.) 

nx\jz  _  {7xyz)x*  _  X* 
^-  2\xy^z~{1xyz)Zy'-^y''^\'' 

xy-\-y        (^  +  i)y  y 

z>-6V_      x\x^-b^)      _     X* 

2x'— 16j;-6_2(x'  — Sir— 3)_2 
^*  3x*-24j:-9~3(j:'-8j:-3)~3' 

8.  The  greatest  common  divisor  of  the  numerator  and  denomina- 
tor, found  by  (105),  is  2x— 3  ;  hence, 

(2x'  —  1x^+l4x—l2)-i-(2x—3)=zx^  —  2x  +  4: 
l4tx'—4f  —  l3x-{-l5)-^(2x—3)  =  2x^-\-x—5 

x* 2a; +  4 

And  we  have  for  the  reduced  fraction,  —^ ,  Ans, 

2x'-\-x—5 

a*c-\-2abc-}-b*c     _       c{a -^  b){a -h  b)      _    c 
-    ^V  a'+3a'6  +  3a6'  +  6»~ (^+^(^6)(a  +  b)~^b '         * 

a'  — 3a'jr  +  3ax'— a;'  _  («— ar)'_  (a— a:)(a— ar)'_a'  — 2ax  +  ic' 
a'— a:'  ~  a^  —  x*       (a—x)(a-j-x)  ~       a  +  x       * 

Ans. 

11.  The  greatest  common  divisor  of  the  numerator  and  denomi- 
nator is  2a^  +  3x  ;  hence, 

(6a' +    1ax—3x^)-^(2a-^3x)  =  3a—x 
(Qa^  +  Uax-\-3x^)^(2a  +  3x)  =  3a+x 

Qa — X 
And  we  have  for  the  reduced  fraction, ,  Ans. 

3a +x 

(67) 


KEDUCTION.  21^ 

12.  The  greatest  common  divisor  of  the  numerator  and  denomi- 
nator is  x'—x'—x-^-l ;  hence, 

^x^^x*—x  ^l)-ir{x^—x^—x+l)=x^  +  l 

(^3;*—X^  —  X^+x)^{x^  —  X^'-X-^l)=zX 

x^  +  1 
And  we  have  for  the  reduced  fraction,  ,  Ans, 

x 

13    (•^  +  yr-2:--/_(ar  +  y)>-(a:'  +  /) 
•   (x  +  yy-x'-f     {x  +  yy-{x'  +  f) 

J^x  +  y){{x-^yY-{x*-x'y  +  x\/--x/-iry')\ 

(x^y)[{x  +  yy-(x'-xy  +  f)] 
__5x'y-h5x^y'-^5xy* 
~  diy 

14.  Removing  the  parentheses  by  multiplication  and  involution, 
"we  have 
(Sx^  —  l)(2x^  —  l)—x\5x^  —  1)  __       Qx*  —  5x^-hl  —  5x*  +  1x*      __ 
(3a:'-l)'  +  (a:'— 3a:)»  ~  9a:*  — 6ar'  +  \+x'—Qx'  +  9a;*  ~" 

_^Nh2^+l (a:'-f  1)'  _      1  . 

a:«4-32:*  +  32:»  +  l  ~  (a^'  +  l)'  ""  a:»  +  l* 

( 125,  page  68.) 

,     2jr*  — 2y*     2(ar— y)(x'  +  a:yH-y')        ,  ,  ,^     , 

4.  -^37^=   ^       ^;^_y    ^^^^  =  2(a:'  +  ary  +  y')>^^> 

^    24a;»— 18a?— 6     „       ^      2ar      6 

6. =3jr— 2 ; 

8a;  8       8 

«      2a; +  6 

=  3a;— 2 — 

8 

=  3a;— 2 ,  Ans, 

4 

^    66a;'  +  126a;-140     „       ^       —14 
8.  ^    .  ^, =8a;— 6  + 


7a;+21  7a;  +  21 

2 

=  8a;— 6 ,  Ant, 

a;  +  3' 

(67-68) 


22 


FRACTIONS. 


10.  '4i:i^=..+v+^4^ 

X  —y  a:  — y 


=;.*+^/+__  y'(— y) 


(  126,  page  69.) 

1.  l+a+-  = ,  Alls. 

2.  26 j—=z 1_,  ^n*. 


a6  +  ar     bah  +  ah-\-x     Qah+x     ^ 

3.  5a  H —  = =  — ^n«. 

6  6  0      ' 

^     ,^     3a  4- 6     126  + 3a +  6     136  + 3a     , 

4.  12+—^—  = = ^Ans. 


^    ^       2j— 5      \bx—{2x—b)      13a-  +  5      . 

6.  3a-9^^^1^^^^^^'-^^-(^"'~^")=-A-      An, 
a  +  3  a  +  3  a  +  S* 

^        .  y'        ar'  — y'  +  y'         a:' 

ar— y  a^— y  ^— y 

8  ,__^'-4a:*4-8_a:'-3x'  +  4-(a;«-4a;'  +  8)        a;'~4 

•  ^+  (a;-2)'     -  (ar~2)'  ~  (^^"  " 

ar  +  2 
a:— 2* 

o      .  .     I.  .  M      «'  +  ^'     a'-6»-(a'  +  6')  26'       , 

9.  a'+a6  +  6' -= ^ ^  — ~,Ans. 

a—b  a  —  b  a  —  b 

T  y-r  y  1-  y -r   j__^,  ^__^,         (l  +  y)(i-y) 

-,  Ans, 

1-y 
(69-70) 


ADDITION.  23 

11    (,    ly     (^-l)'^4^-i)'-('^-i)-^(^-i)(^-i)' 

'    ^  ^.  X  X  X 

(x-iy 


12.  a:»  +  5xy+y*  +  - 


,  Ans, 

X 

21a;y  ar*  +  2xy-25^y4-/  +  21a:y 


i ,  ^n*. 


a;'  — 5ary  +  y'       x*  —  bxrj-\-y 

ADDITION. 
(IJIJO,  page  75.) 
,     Zx     2x     x     63a;4-30j-4-35.r     128a;     , 

^-  T+y+3= Wo =ToF'^"'- 

a     a-\-b     ac  +  ah  +  h*     , 
6         c  he 

^    a*     a*  +  x*     a'  +  a'ar  +  3a'+ 3a;'     , 

3.  —H = — r ,  Ans, 

3       a  +  ar  3(a  +  ar)  ' 

g  +  6     a  —  h     a^  +  2ah +  }}''-{- a"  — 2nh  ■]-})*     la* +  2h* 

6.  The  sum  of  the  entire  quantities  is  6a. 

q  +  3      2a— 5_4a  +  12  +  10a— 25_14a  — 13 

5     "^      4     ~  20  ~"       20      ' 

„                                                                        ^        14a-13      . 
Hence,  6a  h -^ ,  Ans, 

x—2     2ar— 3_5ar'  — 10a;  +  6ar--9_5a;'  — 4a;— 9 
3  5x    ""~  15^  ""        15^        ' 

„  ^       6a;'-4a;-9     , 

Hence,  9j;  +  — ,  Ans, 

15.C 

^       a  2c  c    _a*— ac  +  2ac  +  2c'4-«c— c' 


a+c     a— c     a+c 


( 70-75 ) 


24  FRACTIONS. 

^-  — ^3 +     K. >..»-  +  - 


52:'  52V  10/ 

2xy  — 6y*4-6j;*  +  6y*  +  a;y~-6jr*     2x^y^  ■\- x'^y*     ly  +  x 

9.  "NVe  find,  by  inspection,  that  tlie  least  common  multiple  of  the 
given  denominators  is  {b—c){c—a){a  —  h).  We  must,  therefore, 
multiply  the  numerator  and  denominator  of  the  first  fraction  by 
(a  — ft),  of  the  second  by  (h—c)^  and  of  the  third  by  {c—a)^  to  re- 
duce the  fractions  to  their  least  common  denominator.     Thus, 

a  +  6  a'-6' 


{b-c){c-a)~ (h-c){c—a){a  —  b)  ' 

6  +  c h'-c" 

{c-a){a  —  b)~{b—c){c—a){a—b)  ' 

c-\-a  c^—a* 


(a-b){b-c)~  {h-c)(c-a){a-by 
Taking  the  sum  of  the  numerators,  we  have 

a^-b'  +  b'-c'  +  c'-a*  0 

(b-c)(c-a)(a-b)     -(6_f)(c-a)(a-6) 


0,  Ans, 


10.  By  (109),  we  find  that  the  least  common  multiple  of  the 
given  denominators  is  (a  — i)(a  — 1)(6  +  1).  Hence  the  terms  of  the 
first  fraction  must  be  multiplied  by  6  +  1,  of  the  second  by  —  (a— 1), 
and  of  the  third  by  —(a— 6).     Thus, 

g»_6        _     a^b-b'-\-a'-^b      , 
(a_6)(a-l)~(a-6)(a-l)(6  +  l)  * 

ft'  +  a  — 06'  — a'4-6'  +  a 

(64-i)(6^=(a-6)(a-l)(6  +  l)' 

(l-a)(l  +  6)^(a-6)(a-l)(6  +  l)' 
Taking  the  sum  of  the  numerators,  we  have 
a'b  —  b^-ha'—b-ab'-a'  +  b^+a—a-a^b  0 


=  0, 


(a-6)(a-l)(6+l)  -(a_6)(a-l)(6  +  l) 

(75) 


ADDITION.  25 

11.  The  least  common  denominator  is  {a—b){a—c)[b—c). 

be b'c-bc'  ^ 

{a-b)(a-^~(a  —  b){a-c){b^cy 

ac  ac^  —  aV 

(6_c)(6-a)'^(a-/>)(a-^)^6^)' 

ab  a^b  —  ab* 

{c—a){c—b)  ~  (a  —  l)j{a-c)(b—cy 

^    .  .  a*b-ab*  +  b'c-bc'  +  ac^-a'c 

Their  sum  is  -^ iT-,-n r^, — i r  =  1»  ^^^^' 

ao—ab'  +  b^c  —  bc'-\-ac—a^c 

12.  By  (llO),  we  find  the  least  common  multiple  of  the  given 
denominators  to  be  x*  —  Qx^  +  11-^—0.  Dividing  this  by  the  denomi- 
nators of  the  given  fractions,  the  respective  quotients  are  (a;  — 3), 
(ir— 2),  and  (r— 1).  Multiplying  the  terms  of  the  first  fraction  by 
(ir  — 3),  of  the  second  by  (-^—2),  and  of  the  third  by  (x—l)^  we 
have 

x—3  a-'— 6.C  +  9 

x^  —  3x-i-2~x'  —  Qx*  +  Ux  —  ^  ' 

T—2  a:'  — 4^  +  4 

a:^—4x-{-3~x*—6x*-i-llx-G  ' 

x—l  a:'  — 2a;  +  l 

x^^5x+6~x'  —  Qx^-{-llx^Q' 

Taking  the  sum  of  the  numerators,  we  have 

<r»_6.r4-9  +  a?'~42r-f4  +  ar*  — 2a-  +  l        3a-'— 12a;  +  14 


,  Ans. 


x'—^x^  +  llx—G  x'-6x''-\-llx-6 

13.  The  least  common  multiple  of  the  given  denominators  is  found 
to  be  x'  +  6x^  4- 1  la;  +  6.  Hence,  multiply  the  ternis  of  the  first  frac- 
tion by  (x*  4- 5ar  +  6),  of  the  second  by  (a: +  3),  and  of  the  third  by  1. 

z    _     x^  +  bx^-^-Gx 
ar+l  "^H-ear'  +  lLr+G  ' 

X*  x"  +  3x' 


a:'  +  3.r  +  2     a:'  +  6a:' +  lla;+6' 
a;'  — 2ar'  — 3a: 


«'  +  6a:'  +  llx-f  6     a:' +  6a;'^+ lla:  +  6 
(75) 


S6  FRACTIONS. 

Taking  the  sum  of  the  numerators,  we  have 

X*  -f  52-'  -\-6x  +  x^-\-  3x^  +  x*  —  2x^—Sx_    Sx^  -\-  Qx*  +  Sx 


SUBTRACTION. 

(  130,  page  76.) 

Sx     2x_2lx—l4x  _l2x 
^'  y  "T-        63        -"6F  '  ^'''' 

^    1x     2ar— 1     21ar— (43-— 2)     lYa:  +  2        ' 

x—y     x  +  y  X  —y  x—y 

4.  The  difference  of  the  entire  quantities  is  a,  and  the  difference 
of  the  fractions  is 

11a  — 10     3a— 5_77a  — YO  — (45a  — 75)_32a4-5 
\h  '1~~'  105  ~     105 

XT  .  32a4-5       . 

Hence,  a  H —  ,  Ans, 

a-\-b     a—b_a*  +  2ab-{-b''  —  (a^  —  2ab  +  b^)_   4ab 

6.  We  first  take  the  difference  of  the  fractions. 

x—y        ^•\-y       x^  —  2xy-\-y'^     {x^ ■\-2xy ■\-y^^ _   — 4.ry    _  — 4y 
^■\-xy~x^—xy~'~x{J^^^^  x{x?-y'')     ~  x{x^—y^~  x^-y*' 

Hence,  x-j^-^,  Ans, 


SUBTRAC 


V.  The  difference  of  the  entire  quantities 
of  the  fractions  is 


X     (     X — a\     X     X — a      cx-{-hx — ah 


(x—a\     X 


b     \         c    I     b         c  be  ' 

Hence,  ^X"^"" — ^^- <,  Ans, 

8.  By  (  1 05  ),  we  find  the  greatest  common  divisor  of  the  de- 
nominators to  be  (2:r— 3). 

(2ar'  — ll.r+12)-^(2j:-3)r=:r-4;  (2a;'  +  5ar-12)^(2a;-3)=2:  +  4. 

Hence,  the  least  common  denominator  of  the  fractions  is 
(2a;— 3)(r— 4)(.r  +  4)  ;  and  we  have 

x*-{-x—5     _      x' 4-^-5      _      ir'  +  5ar'— ar— 20 
2j:''-llj;  +  r2~(2j:-3)(.c-4)~(2i:-3)(a;-4)(a:  +  4')'* 

ar'  +  ar-^l     _      ar'+ar-l      _      a;' -  3i:'^-5^-|-4_ 
2x»+5^-12""(2jr-3)(a'  +  4)"~(2i:-3)(a:-4)(ar  +  4')* 

Taking  the  diflference  of  the  numerators,  we  have 

x*+bx*  —  x-20  —  (x*-.'^x'*-br-\-A)  _         83:'4-4a;-24 

(2z-3)(x-4)(2:  +  4)  ~{2x-^)(x-^){x-\-A) 

_  (23;— 3)(4.r  +  8)  _  4a;  +  8 
■"(2x-3)(a:'-16)  ~  ^-16  '  • 

9.  We  may  factor  the  denominators  of  the  given  fractions  by 
inspection.     Thus, 

a'  +  3aft-f  2i'=:(a+    h){a-{-2h)', 
a'  +  5a&  +  66'  =  (a  +  26)(a  +  3&). 

Hence  {a-\-h)(a-{-2b)(a-{-^h)  is  the  least  common  denominator  of 
the  fractions ;  and  we  have  - 

3a +  6  3a'+10a6+36' 

wy 

Hence, 


a*  +  3a6  +  26'     (a  +  J)(aH-26)(a  +  36) ' 
cr  +  76  o' 4-8^6 +  76' 


a'  +  5a6  +  66'     (a  +  6)(a  +  26)(a  +  36) 
(76) 


i58  FRACTIONS. 

3a'-f  10afe4■36'-(a'  +  8(/^>4-T6')_       2a* -\- 2ab  -  4h* 

(a-f  6)(a  +  26)(a  +  36)  ~(a  +  6)(a  +  26)(a  +  36) 

_       2(a  +  26)(a-6) 
~"  (a  +  6)(a  +  26)(a  +  36) 

2(a-fe)  . 

10.  The  denominators  may  be  transformed  as  follows  : 

1ab{a-b)-2(a*-b')  =  {a-b)[1ab-2{a'-\-ab-\-b')]  = 

(a-6)(5a6-2a'-26'); 

3ab{a  +  h)-2(a'  +  b')  =  (a  +  b)[3ab-2{a*-ab  +  b')]=: 

{a  +  b)(5ab-2a*-2b'). 

Hence  the  least  common  denominator  is  (a'  — 6')(5«6  — 2a'  — 26') ; 
and  the  terms  of  the  first  fraction  must  be  multiplied  by  («  +  6),  and 
the  terms  of  the  second  fraction  by  (a  —  b).     We  shall  have 

4q  — 36 4a*  +  a6-36'  ^ 

'ra6(a-6)-2(tf*T6^  ~  (a'-6')(5a6-2a'-26')  ' 

8a  — b  8a'-9a64-6' 


3a6(a4-6)-2(a'+6')  ~  (a'-6')(5a6-2a»-26') ' 
And  the  difference  is 

4a'  +  o6-36'  — (8a'-9a6  +  6»)  iOa6-4a'-46'  2 


(a'-6')(5a6-2a'-26')        -(a«-6')(5a6-2a'-26')~"a'-6'* 

Ans, 


MULTIPLICATION. 

(132,  page  77.) 

1.  Multiply  J-  by  -. 

Canceling  the  common  factor  6,  we  have  - ,  Aits, 

a-\-x  5a  a-{-x  5«  ^      a 

'   "SF^  ^^^"^0^"^  3(7+^)  "^18' 
(76-77) 


MULTIPLICATION.  29^ 

2a  5z  bx      ' 

4/         15y-30      2yx2?/     3(5y— 10)     ^        . 
5y--10  2y  5y-10  2y  ^* 

*    a  +  b^ab-h*  (M^6)  ^6(a-6)~         6        * 

Ans, 

1.  elfz:^^       6a        ^^(«  +  ^)(^-^)^        6^       -3(a  +  ;r),^n.. 
a  2aa:— 2aj'  a  2x(a— a;)       ^         ^' 

.  ar     aft  +  ar        ,  y     ob—y 

aft  +  a;     aft — y     a^h'*-\-ahr. — nhxi  —  xu      . 

__  X  -^=. j^-—- ,  .!»,. 

3a;*— 5a;  7a     _a-(3r— 5)  7a       _ffl(3a;— 5) 

14      ^2a;»-3a;~     7x2      ^  ar(2.r»-3)~2(2a;'-3) 

Sax— 5a      . 
-r—,Ans. 


4a:' -6 


,«    X  —y         X  a        (x  +  y)(x—7/)        x  a 

10.  ^x X ^)^-LlJS — Jlx X =a,An8. 

x         x  +  y     x—y  X  x  +  y     x—y 

,,     4a«-166'  b'l  4(a4-2i)(a  — 26) 

11.  ri— X— - — — — — — =-^ — . — ^^— ^X 


a  — 26         8'*' +  32a6  4- 326'  (a  — 26) 

56 56       _     o6 

2  x4(a4-26)(a  +  26)"~2(a  +  26)~2a  +  46 


,  Ans, 


2a       a  +  6       1        2(a  +  6)  ' 

ax  a'         ^^  a' — x*      a*—b^        a' 

13.  a H — ■.     Hence,    r-x -x =: 

a—x     a  —  x  a-\-b      ax  +  x      a-^x 

(a+ar)(a-a:)^(a  +  6)(a-6)^     «'    ^«'(«-^)    ^^^ 
0  +  6  a"(a  +  a')  a—x  x        ' 

(78) 


SOfc  FRACTIONS. 

a*-z*  ^  a-\-b      a-b^{a'-^x'){a-{-x){a-x)      a  +  b 

^*-  a'^b'     a'-hx'     a-z  (a  +  b)(a-b)  a'+a;''' 

^  a—b  . 

=:a-^x.  Ans, 

a—x 

15.  ^'-^V/'+^^  ^'  ^(•^+^)(»^-^)^^'+^V^  ^^  - 

6c  ft+c      ir— 6  be  b-\-c      x—b 

16.  _J^fczfL_x--^ti±^x^^' 
a'  +  2ac  +  c'     a'  — 2ac  +  c'      ac'x 

^     ^(«-^)       ^      ^(«  +  ^)       x^'"^'^^''~''^=^    Arts. 
(a  +  c)(a4-f)     (a  — r)(a  — c)  acw  «a;' 

^    (a  +  6— c)(rt  — &  +  c)  C4-&— rt 

17.  -5^ ^r ^X;^ ; .-r. — 


a  —  b—c  {c  —  b—a)(b—c—a) 

{a-{-b—c)(a—b+c)  c-\-b—a 

■     (_i)(c  +  6-a)     ^(-l)(a  +  6-c)  x  (-l)(a-6+c) 


(1)(1)(1)         _1_      1,^,, 


(-1)(-1)(-1)      -1 


DIVISION. 
(133,  page  80.) 

I5ab      lOac  _l5ab     {a-\-x){a^x)  _Sh{a-^x) 
a—x     a^—x      a—x  \Oac  2c 

2j:'  — 7  .  a'  _2a;'  — Y      (x  +  a){x  +  a)_ 

x-\-a    '  x^  +  2ax  +  a^~  x  +  a  a'  ~ 

.         ^'-^*       .  ar  +  6_(:r'  +  ft')(.r  +  6)(a:-6)  ^-6_ 
•  ;e*-26a;+6''  •  a:-6~        {^x-b)(x-b)  x  +  b        '^' 

Ans, 

(78-80) 


DIVISION.  :31 

Q :; zzz ^ — — ^ X =^ '■ 

a^—x^    '  a—x     (a—x){a^  +  ax+x'')        x        a!^  +  ax-\-x^* 

Ans, 
14.r— 3      IO2:— 4_14i;— 3        5x5    __70jr— 15 
^-         5~""^'^5~~~~5~"^  10^-4""  lOx-4    »        *• 

9a;'  — 3j;     a:'     ar(9jr— 3)     5  _9a:— 3 
5  5  5  x^  X 

Qx—1     x—l      Qx  —  1         3         18.C-21       , 

I  x-\-x*     2ax+2ax^_x  +  x*  1         _  "^       j 

^^-  "3^"^  7         -~J^'^2a{x+x^)-Q^^'  ^''*- 

a'— a:*  a'  +  ar  +  a:'     a' +a;')(a  — a:)(a'  +  aa;  +  a;') 

a^  —  2ax-^x^  '        a—x  {a—x){a  —  x) 

a  °~  ,    a=a'  +  a;',  ^w?. 

5  5  5  yxy         y 

na—nx     ma — mx     n(a — x)        a  +  b        n 

13.  — r — i — =:-^^^ — r-^X— 7 -=~,Ans, 

f  a-\-b  a-i-o  a-\-o       m{a — x)     m 

X  a        a     x-\-y     x—y     a:'—?/ 

14.  a-^ X =-  X  — -  X ^-= ^  ,  Ans, 

x+y     x—y     I        x  a  x 

3(a;'-l)  ,  /a;  +  l\/g-l\_3(a;  +  l)(-r-l)  2a(a  +  6      _ 

2(a  +  6)    •  V  2a  /U  +  6/  2(a  +  6)         ^  (ar  + l)(a;-l)~" 

3«,  u4w*. 

■^   •   10a6  — 3a'  — 36'"^V6^3a/W"~10a6— 3a'  — 36'^6(3tt  +  6)~' 
(3tf  +  6)(a  +  3^»)     a(6-3a)_3a?>4-a' 
(6-3a)(a^^  ^"6(3a  +  6)~a6-36"     '^**    ' 

^    a'     1     a' 4- a:*      a      1      1     a^—ax  +  x"^     . 

17.  — ,  +  -= T-;  -r— +-= , ;  hence, 

x'^     a        ax^    ^  x^     X     a  ax^ 

a^-\-x*     a^—ax  +  x^     {a-{-x){a*  —  ax+ x*  ax*  a  +  x 

ax*    "^       ^'        =  {ax')(x)  ^a'-aa;Hh^'~    x     ' 

(80) 


a  FRACTIONS. 

a  — 1      6  —  1      c— 1  1  1  1 

18.    +-r-+ 1  =  1 +  l-r  +  l 1=2- 

a  h  c  a  b  c 


REDUCTION   OF   COMPLEX   FORMS. 
(  135,  page  81.) 

2.  Multiplying  both  numerator  and  denominator  by  6c,  we  have 

6 
a  +  - 

c  _ahc-\-b^ 

c      ahc  +  c" 

3.  Multiplying  both  numerator  and  denominator  by   a'6V,  wo 
have 

^+^ 

6c'     a'c_a*6-f6*c 

6'c     ac' 

4.  Multiplying  both  numerator  and  denominator  by  mn,  we  have 

x—\__x-\-\ 

m  n    __nx— n— mar— m_ar(n— m)  — (n  +  wi) 

arH-1      a:— 1      nz+n-^mx—m     ar(n  +  m)  +  (n— w)  ' 

Til  n 

6.  Multiplying  both  numerator  and  denominator  by  a6,  we  have 

^±1-2  +  ^^ 

_b a    _a«  +  a  — 2a6+6'—6_(a— 6)* +  (a  — 6)  _a— 6  +  1 

g-l  6  +  l~a'— a-2a64-6'  +  6~(a  — 6)'-(a— 6y~a— 6-1  * 

.  6  a  Ans, 

6.  Multiplying  both  numerator  and  denominator  by  a6c,  we  have 


^+A+4 


.3   .    A«   I   ^a 


ah     ac      be      a^6''  +  aV+6V 
c       6       a 


r:, »  >4"«- 


(80-82) 


REDUCTION  OF  COMPLEX  FORMS.  33 

7.  Multiplying  both  numerator  and  denominator  by  (c4-rf)(c— c?), 
or  its  equal  c^—cT,  we  have 
a+b     a—b 


c-\-d     c—d      ac-\-bc—ad—bd-\-ac—bc-\-ad  —  bd     2ac  —  2bd 


a-\-b     a—b     ac  +  bc-{-ad  +  bd-\-ac—bc—ad-{-bd     2ac-\-2bd 

c—d     c-^d  ac—bd 

— — -  ,  Ans, 
ac-\-bd 

8.  Multiplying  both  numerator  and  denominator  by 

(a*  — 6')  (a' +  6'),  or  its  equal  a*  —  b\  we  have 

a*  +  b'    «*-y 

a'-&'    (^^IT' a* -f  2a'6'  +  b*  -  {a*  -  2a'b'  +  b*) 

a  +  b  a-b  ~a*  +  2a'b  +  2a'b'  +  2ab'-^b*-{a'-2a'b+2a'b'—2ab'-\-b*) 
a—b^  a+b  4a'6'      _    ^^        a 

9.  The  least  common   multiple  of  the  denominators  of  the  frac- 
tional parts  is  {x*  —  1  )(y'  —  1 ) .     Therefore, 


+  - 


x—l^x-{-l_xf  —  x  +  i/*  —  l-\-xi/*—x-'y*-^l 
1  _l_""ar'y-y+ic'  — l+xV— y-ar'  +  l 

y  — l"^yf  1  22ry'-2^     xy^—x_x(y^  —  \ 


2x*y-2y     xhj-y     y\x'-lf 


,0.  i±i+i+i_£±l_l+I=i+l+i+l_i_l_i4 

aoca  a6ca 

Hence  the  numerator  =--f- -, 

abed 

Substituting  this   expression   for  the  numerator  in  the  complex 

fraction,  and  multiplying  by  abcd{c-\-d){a-^b)^  we  have 

1     1__1     1 

a     b     c~~d     {c-\-d)(a-\-b){bcd  +  acd—ahd—abc) 

cd         ab  ~       abcd{a  +  b)cd~abcd(c+d)ab 


c^d     a+b 

(c  +  d){a-\-b)(bcd  +  acd—abd—abc) 

abcd{bcd  +  acd—abd—abc) 

_(c  +  d)(a  +  b) 

aUd '^'''' 

(82) 


34 


SIMPLE  EQUATIONS. 


1.  Given, 
fcultiplying  by  12, 

2.  Given, 
multiplying  by  42, 

3.  Given, 


SIMPLE    EQUATIONS. 

(  151,  page  88.) 

X     9x     3x     ,^ 
_4._--=10; 

6j:  +  8jr— 9x=:120,  Ans, 
3jr_2j:  +  3_a:— 5  ^ 
18j— 6jr— 9  =  2a;— 10,  Ans, 


-^  +  -i 


d 


multiplying  by  (j;*— a'),         axi-a*  +  cx—ca=dy  Ans, 

x—a     2x—Sa     x  +  ac 


4.  Given, 
multiplying  by  aV", 

5.  Given,  , 

*  8c  10a  4ac     ' 

multiplying  by  40ac,  ba^x—bahx—Ac^x  +  4:acx=10hx—  lOcar,  ^n*. 


c  ac'  a'     * 

aVj;— a*c— 2aa;  +  3a'=:c'a;  +  ac*,  .4n». 

aa; — hx     ex— ax     hx — ex 


6.     Given, 


5x     Zx     Z—x     bx—2 


12     10       24  20 

multiplying  by  240,      100a;— 45ar  + 30  — 10^;— 60a; +  24=480,  Ans, 


1.     Given, 
multiplying  by  abcx, 


1  a        b         c 

abc     box     acx     abx^ 
a;=a'  +  6'  +  c',  Ans. 

(88) 


REDUCTION.  35 

REDUCTION  OF  SIMPLE  EQUATIONS. 
(156,  page  92.) 


1.    Given, 

1x-l6  =  Sx-4t', 

transposing  and  uniting, 
whence,  by  division, 

4:rri:12; 
x=3,  Ans, 

2.     Given, 

3a;+9  =  5a;  +  l  ; 

transposing  and  uniting, 
dividing  by  —2, 

-2a:=:-8; 
x=4^  Ans. 

8.     Given, 

4x  +  1=x-h2l  —  3-{-x; 

transposing  and  uniting, 

2x=n  ; 
x=5^j  Ans, 

4.     Given, 

5x  +  l6=x-[-52\ 

transposing  and  uniting, 

4a:=36; 
x=:9,  Ans, 

6.     Given, 

5ax—c  =  b  —  3ax*^ 

transposing  and  uniting. 

8ax=b  +  c; 

dividing  by  8a, 

b-\-c       . 
^-   8a    »^"*- 

6.     Given, 

ax-\-b=:9x-{-c  \ 

transposing  and  factoring. 

x{a-9)=c-b', 

dividing  by  (a— 9), 

^-^_^,Ans, 

7.     Given, 

H--^ 

clearing  of  fractions, 

3ar  +  2:r=  10x12; 

uniting, 
dividing  by  5, 

5ar=10xl2; 
x=   2x12; 
ir=:24,  Ans, 

(92) 

da 


SIMPLE    EQUATIONS. 


8.     Given, 

clearing  of  fractions, 
transposing  and  uniting, 


9.  Given, 

clearing  of  fractions, 
transposing  and  uniting, 
"whence, 

10.  Given, 

clearing  of  fractions, 
transposing  and  uniting, 

11.  Given, 

clearing  of  fractions, 
transposing  and  uniting, 


12.  Given, 

clearing  of  fractions, 
transposing  and  uniting, 
dividing  by  (—18), 

13.  Given, 

clearing  of  fractions, 
transposing  and  uniting, 
whence,  by  division, 

14.  Given, 

clearing  of  fractions, 
transposing  and  uniting. 


3x     X     ^^ 

Y=l  +  '*' 

6x=x-}-96  ; 
5x=96', 
x=19i,  Ans, 

3a;  +  5_15a:— 1  ^ 
~~2      ~       8       ' 
12x4-20=15x— 1; 
_3j:=-21; 
ar='7,  Ans, 

x+l      Sx—5     9x  ^ 
3     "^      5      "^10*' 
10jr+10  +  18x  — 30=:2'7a;; 

x=20,  Ans. 

2x-\-l      'iX  —  l5_l1x-\-Z     3^ 
'~2      "^       6       ~       8     '~2* 
40jr  +  20  4-562r— 120  =  85a:H-15  — 60  ; 
lla:=55; 
x  =  5y  Ans. 

X     X     5x     5x     dx 

-4--4- — =r 1 18  : 

2^3^12      7^4  ' 

42arH-282:  +  352:  =  60a:  +  63^  — 18  x  84  ; 
—  18x=  — 18x84; 
a?=84,  Ans. 

I7ar—12_6j:--16__10a;— 3_6a;— 7  ^ 
3  4  6  2~* 

68jr— 48  — 15j:  +  48  — 20a:  +  6  =  36ar— 42  ; 
-3x=— 48; 
a;  =16,  Ans, 


21 


3X-11     5x—5     9l  —  1x 


16  8  2 

336+3a?— 11  =  lOjr— 10  +  776  — 56a;; 
49ar=441  ; 
x=9,  AnM. 
(92) 


15.  Given, 
multiplying  by  21, 

transposing  and  uniting, 

clearing  of  fractions, 

whence, 

and 

16.  Given, 
multiplying  by  36, 

dropping  9j?, 

clearing  of  fractions, 
transposing  and  uniting, 
whence. 


17.  Given, 

,         .       20jr  ,  36 
dropping  _+- 

clearing  of  fractions, 
transposing  and  uniting. 
Whence, 

1 8.  Given, 

clearing  of  fractions, 
transposing,  <fec., 
dividing  by  (—3), 

19.  Given, 

clearing  of  fractions, 
transposing,  <fec., 
whence^ 


REDUCTION. 

Ix+IG 


37 


x+8 


21 


4^-11     3* 


H    .  ,^     21^  +  168      ^ 


4X-11 


16: 


21a!  +  168 


4^—11    ' 
6ix—ll6  =  2lx+lQS; 
43x=3U; 
x=8j  Ans» 

92:4-20_4a;  — 12     x^ 
36  bx-4:  "^4' 

^       ^^     144^-432  .  ^ 
0.r  +  2O  =  — :: : \-9x; 


20  = 


5x  — 4 
144a;— 432 


5x-4:      ' 
lOOar— 80  =  144^?— 432; 
— 44ir=  — 352; 
x=8y  Ans, 

20ar     36     5a;  +  20_4jr      86^ 
25      25     9x-16~'"5"     25* 
5a;  +  20_50_ 
9ar-16~25""    ' 
6x+20  =  18ar— 32; 
—  13a:=— 52; 
ar=4,  Ans. 


3x     x—\ 


=  Qx- 


20ar4-13 


4  2  4        ' 

3j;_2a;  +  2  =  24a;— 20a;--13  ; 
— 3x=  — 15; 
a; =5,  Ans, 

ar— 3     X     ^^     a:  +  19 

8ar-.9  +  2a;=120— 3a;— 57  ; 
8a;=72; 
a;=9,  Ana, 
(92-93) 


38 


SIMPLE   EQUATIONS. 


20.  Given, 

clearing  of  fractions, 
transposing,  <kc^ 
whence, 

21.  Given, 

clearing  of  fractions, 
transposing,  &c., 
dividing  by  (  —  11), 

22.  Given, 

dropping  —29, 

,     .       6ar-12     ^  ^ 
reducing 1  to 

X  —  iS 

the  form  of  a  fraction, 

clearing  of  fractions, 
dropping  5ar',  transposing 
and  uniting. 


23.     Given, 


removing  the  parenthesis, 

clearing  of  fractions, 
transposing  and  uniting. 


24.     Given, 

removing  the  parenthesis, 

clearing,  of  fractions, 
transposing  and  uniting, 
dividing  by  (—175), 


x  +  1     a;-f2 


16 


ar  +  3 


2      '      3  4     ' 

6j;  +  6  +  4x  +  8==192  — 3a:— 9  ; 
13^=169  ; 
a;=13,  Ans. 

x  +  3     ,^      12a;  +  26 
2._-_+i5=— ^— ; 

3O2;— 5jr— 15  +  225  =  36a:  +  'r8; 
—  llx=  — 132; 
a;=12,  Ans, 

6X-12 


5j;4-5 


29: 


X  +  2  X-r2 

6ar  +  5     6j;— 12 


-30 


a:+3L      x—2 

x  +  2~~  x—2    * 

5j;'_5j;_10  =  5j:'  — 20 ; 
— 5ar=  — 10; 
x=2f  Ans. 
1x-{-9     /       2ar— 1 

X- 


1 ; 


;  +  9     /       2ar-l\ 

1— r-^=^' 


1x  +  9 


■x  +  - 


9 
2x 


■=1 


1  +  9x 


63ar  +  81  —  36j:  +  8j;— 4  =  252; 
35x=l75; 
x=5f  Ans. 

63  +  81ar— 36-f  8  — 4a:=:252a;; 
-1753;=— 35; 
.  a:=i,  Ans, 
(93) 


REDUCTION. 


^ 


25.     Given, 


x-{-l     x-\-2_x- 


3     x-4 


2  3  4  6 

clearing  of  fractions,  6j;+6H-4ar-f  8  =  32:— 9  +  2a;— 8  +  36  ; 

transposing  and  uniting,  5j?=5  ;  and  a;=l,  Ans. 


26.     Given, 

2  +  3+4  +  5  =  "' 

clearing  of  fractions, 

30x  +  20x  +  15x+12x=11  xQtO; 

uniting, 

11x=11x60; 

whence,  by  division. 

a;  =  60,  Ans, 

27.     Given, 

I+H=-' 

clearing  of  fractions. 

6a;  +  4a;  +  3a;=130x  12; 

uniting, 

132r=130xl2; 

a:=10xl2; 

ar=120,  Ans. 

28.     Given, 

i+i+^=-= 

clearing  of  fractions. 

6a;  +  2a;4-a:  =  90xl2; 

9x  =  90xl2; 

a:=10xl2; 

ic=120,  Ans, 

29.     Given, 

M+r-' 

clearing  of  fractions. 

21aJ4-14a:+6d;=82x42; 

uniting. 

4l2;=82x42; 

a:=2x  42  =  82,  Ans, 

30.     Given, 

l+?+^+^+^=«««' 

clearing  of  fractions. 

84a;  +  60ar  +  35a;  +  21ar  + 202^  =  660  x  420  ; 

uniting, 

220a;=660x420; 

dividing  by  220, 

a:=3x420; 

whence. 

ar=1260,  Ans, 

(93) 


40  SIMPLE   EQUATIONS. 

31.     Given,  a'x  +  2ac— c'ar=a'  +  c' ; 

transposing,  a'x— c'a:=a'  — 2ac  +  c' : 

dividing  by  (a*—c^)^  and  factoring,  x=^  —c)[a—c)  ^ 

(a—c)(a  +  c)  ' 


or. 


X———  ,  Ans, 
a-\-c 


32.     Given,  Ahx—2a=Zab  —  Qh''x 

transposing,  66'a;  +  46x= 3a6  +  2a ; 

Zab  +  2a ^ 
W+Ab' 
a 
26 


whence, 


reducing,  xz^~,  Am, 


33.  Given,  a{x—h)-\-h{x—c)-\-c{x  —  a)=0\ 

removing  parentheses,    ax—ab  +  hx  —  hc-\-cx—ac=0\ 

transposing,  ax  +  bx  +  cx=ab-\-bc-\-ac*, 

factoring,  x(a  +  6  +  c) =a6  -f  6c  +  ac  ; 

,  ab  +  bc-\-ac     ^ 

whence,  ar= Ans. 

a+b+c    ' 

34.  Given,  a'(ar— 1)  4-am(a:— 2)=m'; 
removing  parentheses,  a*x—a*-\-amx—2am=m* ; 
transposing  and  factoring,  a[a-\-in)x=a*  +  2am -\-m* ; 

whence,  x= ,  Ans. 

a 

So,     Given,  ax-{-cx  +  x=b-\ ; 

c 

clearing  of  fractions,  acx  +  c*x-\-cx=bc-\-b—ax; 

transposing,  acx-\-ax-^c^x-{-cx=bc  +  b\ 

factoring,  ax{c  +  l)-\-cx{c-\-l)  =  b(c  +  l) ; 

dividing  by  (c  + 1 ),  ax-{-cx=b] 

whence,  x= ,  Ans, 

*  a-[-c* 

(93-94) 


REDUCTION.  41 

^.  a-\-x     c  —  x     a 

36.     Given,  __  +  __=_; 

cleari  ng  of  fractions,  ad-\-dx-\-bc— bx = ad ; 

transposing,  —bx  +  dx=--bc; 

or  by  (150,  III.,)  b.e^dx=bc', 

whence,  x=j-—^,Ans, 


X  X  X  X 

37.    Given,  -  +  - — — ---— — -=1. 

a— 1      6—1     a+\     6+1 

.  X  X  2x  X  X  2x 

Observing  that      -— — r-r=i — 7»  ^^^  I — \ — TTT=T5 — 7» 

^  o— 1     a  +  1     a'  — 1  6—1      6  +  1      6'  — 1 

we  have  ^rri+fti^i^^* 

clearing  of  fractions,      26'x-2ar+2a'x-2a:=(a»-l)(6'-l) ; 
uniting  and  factoring,  2(6'— 2  +  a').ci=(a'-l)(6'  — 1) ; 


(.«-l)(6»-i) 


«— 1         X           \  2 

38.    Given,  rH — —r= -  + 


c— 1     c  +  1      c— 1     (c  — 1)" 

car— a;  2 

multiplying  by  (c— 1),  *~"^  +  7+Y-^ +^Z:i ' 

car— ar        2 

uniting,  :r— 2H —- = ; 

*"  c  +  1      c— 1  ' 

clearing  of  fractions,        c'a;— ar— 2c'  +  2+c'a;— 2ca;  +  ar=2c  +  2  ; 

transposing  and  uniting,  2c'ar— 2ca; = 2c'  +  2c  ; 

dividing  by  2c,  and  factoring,  (c— l)a:=:c+l  ; 

whence,  x— -,  Ans, 

c — 1 

XXX 

39.    Given,  -+T^ — =a6  +  ac  +  6c; 

a     6     c 

clearing  of  fractions,  a;(6c  +  ac-\-  ab)  —  {ab-\-ac  +  6c)a6c ; 

whence,  x=abc,  Ans. 

(94) 


SIMPLE    EQUATIONS. 

X — h — c     T — a — c     X — a  —  h 


+  -6— +  — 


3; 


42 

40.     Given, 

clearing  of  fractions, 

hex —  (6  ■{■€)})€ -Y  acx—  (a  +  c)ac  ■i-abx-—(a.  +  h)ab=iSahc  ; 
Transposing, 

bcx-^acx  +  abx=(b  +  c)bc  +  (a  +  c)ac  +  (a  +  i)a6  +  Sabc. 

Removing  the  parentheses,  and  arranging  the  second  member  of 
the  equation  in  three  terms  with  reference  to  the  coefficient  of  ar, 
we  have 

(be  +  ac  +  ab)xz=  (abc  +  tt'c+a'6)  +  (b^c  +  abc  +  ab^)  +  [be''  +  ac*  +  abc\ 
dividing  by  {be  -\-ac-\-  a6),         a*= a  +  6  +  c,  Ans. 


41.  Given, 
transposing, 
uniting, 
dividing  by  .875, 

42.  Given, 
transposing  and  uniting, 
dividing  by  2.844, 

43.  Given,  ^  ^  . 

'  2.8  4  .7 

clearing  of  fractions,  2.4a;— .12  +  3.22j;— 2.52  =  2.56a;— .192  ; 

transposing  and  uniting  3.06a; =2.448  ; 

x=,8,  Ans. 


1.25X— 6.125 +  .25x=.625a:; 

1.25a;  +  .25a;— .625a;  =  6.125  ; 

.875a:  =  6.125; 

a: =7,  Ans 

3.164a:— 4.266  =  . 24a; +  . 08a: ; 
ra:  =  4.266; 
a;=1.5,  Ans. 


2.4a;— .12     4.6a:— 3.6     .64.r— .048 


PROBLEMS 

PRODUCING    EQUATIONS    WHICH    CONTAIN    ONE    UNKNOWN    QUANTITY. 

(160,  page  97.) 

1.  Let  X  represent  the  number  ;  then  by  the  conditions, 

(a;-6)ll  =  121; 
whence,  a:— 6  =  11  ; 

a;=l7,  Ans. 
(94-97) 


PROBLEMS.  ,  '43 

2.  Let  anytime  past ;  then 

20— a:=time  to  come. 
Now  by  the  conditions  of  )  a;_20— a?  ^ 

the  problem,  we  have     i  3         2' 

whence,  2j:=:60— 3j;; 

5a: =60,  and  2-=  12,  Ans, 

3.  Let  X  represent  the  number  ;  then  by  the  conditions, 

XXX 

^  +  2-^3  +  4^^^^' 
whence,  25a;=250  x  12  ; 

a;:=  10  X  12  =  120,  ^7W. 

4.  Let  if = one  part ;  then 

7  7 — a; = the  oth  er  part ; 
and  by  the  conditions  of)  x     'J^'7— a;_ 

the  problem,  we  have  j  7         3"' 

8ar  +  539  — 7^=315; 

_4a:=_224; 
whence,  x—56y  one  part; 

and  77— a:= 21,  th»  other  part. 

5.  Let  a; = the  greater  number;  then 

75— a;=  the  lesser  number; 

and  by  the  conditions  of  )  ,  . x 

the  problem,  we  have  j  ^  '~3' 

3^—225  +3a:r=a:; 

5ar=225; 
whence,  a;=r45,  the  greater  ; 

and  75— a;=30,  the  less. 

6.  Let  X  represent  the  amount  of  money  at  first ;  then  by  the 
conditions,  we  have 

^     ^      «« 

20a:— 5a;— 4jr  =  66x20; 
lla:  =  66x20  ; 
whence,  x—   6  x  20  =  120,  ^w*. 

(97) 


44 


SIMPLE   EQUATIONS. 


7.  Let  30jr  represent  my  money  at  first ;  then 

30.i: —  =  20j:=: remainder  after  the  firet  payment; 

3 

20x J- =  152:=: remainder  after  the  second  payment; 

15z 

whence,        loz =12a;=24  ; 

5 

x=2; 
30ar=60,  Ans. 


8.  Let  X  represent  the  number  ;  then  by  the  conditions, 


^(^-5)  =  40; 

2ar-10=:120; 

ar=65,  Ans, 

9.  Let 

arr^price  of  the  horse  ;  then 

200— ar=:price  of  the  chaise  ; 

and  by  the  conditions, 

1=^(200-.) ; 

3jr=400-2ar; 

a:=80,  the  price  of  the  horse. 

200— ar=120,  the  price  of  the  chaise. 

10.  Let 

a?=the  greater  part ;   then 

48— ar= the  lesser  part; 

and  by  conditions, 

6^     4     -^' 

4a;  +  288  — 6a;=216; 

whence, 

ar= 36,  the  greater; 

and 

48— a;=12,  the  lesser. 

11.  Let  X  represent  the  value  of  the  estate;   then  by  the  con- 
ditions, .         *  ; 

(97-98) 


PROBLEMS.  45 


X 

--f  200= what  tlie  first  received, 
4 

/»• 

-  +  340= what  the  second  received, 
5 

/p 

-  +  300  =  what  the  third  received, 
6 

/p 

-  +  400  =  what  the  fourth  received, 
8 

whence,  T^f +  ^  +  ^  +  124Q=ar ; 

4     o     o      o 

30ar-f  24«+20ar+15a;+1240  x  120  =  120x  ; 

31j;=1240x120  ; 

ar=40  X  120  =  4800,  ^n«. 

12.  Let  X  represent  the  number  ;  then  by  the  conditions, 

whence,  lOi:  — 910  =  3  j; 

7^=910,  and  «=130,  Ans, 

13.  Let  ar=  A  s  stock  ;  then 

3j=  B's  stock  ; 
12ar=  C's  stock ; 
5^=  D's  stock. 
By  the  conditions,  21ar=  V3500,  and  ar=$3500,  Ans, 

15.  Let  *lx—  A's  share;  then  9^;=  B's  share;  and  by  the  con- 
ditions, 

7^  +  9^=  $2000; 
whence,  '7a:=$875,  A's  share  ;  9x=$1125,  B's  share. 

16.  Let  Zx=  the  rye ;  then  ^x=  the  oats,  and  5x—  the  peas. 
By  the  conditions, 

3i:  +  4a;  +  5a:=72  ; 
whence,  ar=6  ;  3jr=18;  ^xz=z2A^  and  5ar=30; 

therefore,  Rye,  18  bushels;  oats,  24  bushels;  peas,  30  bushels, -4w5, 

(98) 


SIMPLE   EQUATIONS. 


17.  Let  6j*=  the  quantity  of  wine  drawn  from  one  cask;  then 
1x=z  the  quantity  of  wine  drawn  from  the  other. 
By  the  conditions, 


whence, 


6x 
'7x—16=—=3x 
2 

x=4,  7.r  =  28,  and  6a;=24. 


18.  Let 
then, 
whence, 


5x=  less  part,  and  a=204 ; 
a—5x=  greater  part ; 
a— Yjr  =  20j— f(a  — 5.r)  ; 
204j;=10a  =  iox204; 
a:=10; 
5.r=50,  the  lesser; 
204  —  5^=154,  the  greater. 


19.  Let 


Now  by  the  conditions, 


whence, 
and, 


Sx=  the  price  of  the  horse  ;  then 
a—8x=  the  price  of  the  chaise. 

2a  —  16x—Sx=2ix—^{a  —  Sx)  ; 
148  — 133ar=:1682:— 5a-+-40jr, 
341ar=19a=19x341  ; 
ar=19; 
8ar=152,  horse, 
341  — 8j:=189,  chaise. 


21.  Put  I560  =  a  ;  then  9560  =  a  +  2000. 

Let  x=  each  private's  share  ;  then 

2lx=  what  all  the  privates  received ; 
2*Jx-{-a=  the  prize. 
By  conditions  of  the  problem,  we  have 

27a: +  a=:25j:  +  a  + 2000; 
2a;=2000;  a:=1000; 


27a;  +  7560: 


34560,  Ans. 


22.  As  he  annually  increases  his  unexpended  capital  by  one  third 
of  itself,  four  thirds  of  this  unexpended  part  will  represent  his  capi- 
tal at  the  end  of  each  year. 

(99-100) 


PROBLEMS. 


47. 


Let 


therefore, 


flr=  his  original  stock;  put  1000=ff;  then 

4  .         .      4x  —  4a_  stock  at  the  end  of 
3^  3       ~       the  first  year ; 

4/4x—4a       \     16j?— 28.c_  stockat  the  endof 
3\      3  /~        9  the  second  year  J 

4/16ar— 28a       \     64:c— 148a_  stock  at  the  end  of 
3\        9  /~        27        ~       the  third  year; 


27 
64a;  — 148a 


2x 


27 

64ar— 148a  =  54a;; 
10a:=148a=148xl000; 
a;=:  14800,  Ans, 


23.  Puta=99.     Let 
Now  by  the  conditions, 


whence, 


a'zsthe  time  past ;   then 
a— a? = the  time  to  come. 

2a:     4.  , 

10a;=12a  — 12a;; 
22a;=12a  =  12x99  ; 
a;=6x9=:54; 
99— a;=:  45,  Ans. 


24.  Let 


6a;=the  amount  of  gunpowder;  then, 
4a; +  10= the  nitre; 
ar— 4|=the  sulphur; 
4a;+10 


By  the  conditions. 


whence, 


2=the  charcoal. 

4a;— 10 


4a;  +  10+a;-44  + 


4a;— 10 


2  =  6x; 


1  •^-     2' 

8a;  +  20  — 14a;=— 49; 

6a;=69,  ^w*. 


(100) 


48 


SIMPLE   EQUATIONS. 


25.  Put  a= 183.     Let 


By  the  conditions, 


whence, 


a:=what  the  first  received  ;  then, 


a— x^what  the  second  received. 

40jr=21a  — 21J-; 

21  xl83; 


a;=:21  x3  = 
183— ar^  120  ) 


26.  Let 

By  the  conditions, 
whence. 


a:=the  greater  part;  then, 
68— :r=the  lesser  part. 
84— ar=3[40-(68-a-)]; 
84— ar=120  — 204  +  3J:; 
4z=168, 

68  — a;=26  ) 


27.  Let 


l/2jr     3j-\ 

3(t+t) 


2z=  distance  from  A  to  B  ;  then 
32r=  distance  from  C  to  D ; 

:—  —  distance  from  B  to  C. 


By  the  conditions  of  the  problem, 
5ar+y=34; 

1Vj:=34x3; 


Whence, 


x=6. 

2x=12,  distance  from  A  to  B ; 
3x=:18,  distance  from  C  to  D; 
^x=z  4,  distance  from  B  to  C. 


28.  Let        3ar=the  number  of  sheep  at  first;  then 

2a;— 6=the  number  after  the  first  plunder, 
2x— 6  — (ar— 3)  — 10  =  2; 

a:=:lo;  3x=^45,  Ans, 


Whence, 


29.  Observe  that   for  every  vessel  broken,  he  lost  12   cents,- 
3  cents  fee,  and  9  cents  forfeiture. 

(100) 


PROBLEMS. 


49 


Let 


30.  Let 


By  the  conditions, 


x=  the  number  he  broke  ;  then 
300  — 12j?=:240; 

x=5j  Ans. 

x=  hia  indebtedness  to  A ;  then 
2x=  his  indebtedness  to  B ; 
6x=  his  indebtedness  to  C. 
9j:=270  ;  and  x  =  30. 


31.  Let  ()Xz=    A's  share,  and  9x=  B's  share  ;  then 

20a:=C's  share,  and  433ar=:D's  share. 
By  the  conditions,     l8^z=3lo  ;  whence, 

Gx=24,  9x=S6,  20jr=80,  433a:=:l75. 

32.  Let  5 j:=: his  money  at  first;   then 

4x4- 4  =  what  he  had  after  first  losing  and  winning; 
3ar  +  34-3  =  what  he  had  after  second  losing  and  winning. 
As  he  loses  }  of  this,  he  must  have  f  of  it  left ;  therefore 

i(3a:-f6)  =  20; 
15a;  4-30  =  120; 
whence,  15a:=90,  and  5a:=30,  Ans. 

33.  Let  3.c=:his  income  ;  then 

2j;=his  family  expenses. 
The  remainder  of  his  income  is  a;,  |  of  which  he  spends  in  improve- 
ments, leaving  |  of  a;  to  lay  by ;  whence, 

X 

-  =  70  ;  and  3ar=630,  Ans. 


34.  Let 

By  the  conditions, 
whence, 

35.  Let 

By  the  conditions, 


orrzithe  lesser  part ;  then 
60— a:=:the  greater  part. 
x{eO-x)  =  dx''; 
60~a:=3a:; 

4ar=60,  and  ar^lS,  A7is. 

a?=the  value  of  the  saddle  ;  then 
8jr=the  value  of  the  horse. 
9a:=90  ;  whence  a;=10,  Ans. 

(100) 


50 

36.  Let 
By  the  conditions, 


SIMPLE   EQUATIONS. 


ar=:what  one  receives  ;  then 
10^=  what  the  other  receives. 
Il2r=462  ;  whence  a*=42,  Ans. 


37.  Let  armrent  of  the  estate  last  year ;  then  by  the  conditions, 

8x 
^  +  100  =  ''''' 

108x=  189000,  and  x=ll50,  Ans, 

ar=the  greater ;  then 
840-x=the  less. 

3^— 2520  +  3^=a? ; 

5irr=2520,  and  x=504,  Ans, 


whence, 
38.  Let 

By  the  conditions. 


whence, 

39.  Let  «=hi8  income ;  then  by  the  conditions, 
.-g  +  10o)=|  +  35; 


10a?— 2a:— 1000  =  5ar  +  350 


whence. 

3a:=1350,  and  a; =450,  Ah 

40.  Let 

a;=A's  part;  then 

100+ir=B's  part. 
370+a;=C's  part. 

By  the  conditions, 
whence, 

3a:+470  =  1520; 

a;=350,  A's  part. 

"•Let      )(^.:,^. 

By  the  conditions, 
whence. 

7a; = the  income  ;  then 
3"= A's  annual  debt; 

— =what  B  saves  annually. 
2(y)  =  2a:  +  32; 

a;=40; 

7a; =2 80,  Ans, 

(101-10^) 

PROBLEMS. 


51 


42.  A^s  rate  of  travel  is  |  miles  per  hour ; 

B's  rate  of  travel  is  f  miles  per  Lour; 

66 

— -=the  distance  between  A  and  B,  when  B  sets  out. 
5 

Let  «=the  number  of  hours  ;  then  by  the  conditions, 

6x_*ix     56 

25a:=2U  +  168; 

whence,  '  0^=42,  number  of  hours; 

*bx 
-—=^0^  number  of  miles. 


43.  Let  ar=  the  number  of  days ;    then  both  working  together 
will  do  ~  of  the  work  in  one  day.     But  A  does  |  of  the  work  in 

X  J  • 

one  day,  and  B  j*j  ;     therefore, 

1     J^_l. 
8'*"l2~ar' 

X        X 


whence, 


8     12 
3ar  +  2a;=24,  and  a;=4f,  Ans, 


44.  Let 


x=.  the  distance  ;  then 
-=  hours  he  rides ; 


By  the  conditions, 
whence. 


-=:  hours  he  walks. 
4 

X      X      ^ 

8j:=48,  and  a;=16,  Ans, 


45.  Let  3a:  =  the  time  in  which  C  can  dig  the  trench  ; 

then  2ar=  the  time  in  which  B  can  dig  the  trench ; 

and  x=.  the  time  in  which  A  can  dig  the  trench. 

(102) 


52  SIMPLE   EQUATIONS. 

Now  in  one  day  A  can  dier  -  of  it,  B  -—  of  it,  and  C  —  of  it ; 

but  all  together  can  dig  -  of  it  in  one  day.     Hence, 
>^^  , ^  1111 


^  7        J  ar=ll.     As  time, 

.-40=^ /S  '^^  2x=22,     B'stirae, 

^j.'^<?..-.'^^>t-^  g^^33      c'stime. 

/  46.  Let  x=  the  distance  between  A   and  B.     Since  C   and   B 
A        travel  at  the  respective  rates  of  5  and  4  miles  an  hour,  when  they 

5  5x 

have  met,  C  has  traveled  --  of  the  whole  distance,  or  —  miles.     In 
y  y 

3  3        bx 

the   meantime,  A,  traveling  -  as  fast  as  C,  has  traveled  -  of  — ,  or 

5  5        9 

X  5x     X     2jj 

-  miles.     Hence,— — — ,  the  distance  between  A  and  C  when 

3  9      3      9 

C  turns  back. 

5  5 

Now  C  in  traveling  back  to  meet  A,  goes  -  of  this  distance,  or  - 

8  8 

of  — = —   miles.     But  the   whole   distance    traveled    by  C  is  60 
9      36  "^ 

miles;  hence,  5x     5x     ^^ 

¥+3-6  =  '*°' 

x=l2y  Ans. 

47.  Let  x=  the  valueof  one  sheep;  then 

l2x-\-92x  —  164x=  the  value  of  the  flock. 
By  the  conditions, 

92*— |35  =  72:r+$35, 

a-=$3i;  whence,  164a;=$574,^n*. 

48.  Let  x=  the  rate  of  the  current  per  hour.  As  the  current 
will  retard  the  boat  by  its  whole  velocity  in  going  up  the  river,  and 
accelerate  it  the  same  in  going  down,  12— x=  actual  rate  of  row- 
ing up  stream,  and  12  +a;=  actual  rate  of  rowing  down  stream.     By 

the  conditions, 

7(12-ar)  =  5(12+ar); 

84  — 7.r=:G0  4-5x; 

12i:  =  24; 

whence,  ar=  2,  Ans. 

(102) 


TWO   UNKNOWN   QUANTITIES.  5& 

SIMPLE    EQUATIONS 

CONTAINING    TWO    UNKNOWN    QUANTITIES. 

(irO,  page  109.) 

1.     Given  i    ^^  +  ^^=   ^^'  (^) 

^'  •il2;r  + 7^=100;  (2) 

multiplying  equation  (1)  by  3,  and  equation  (2)  by  2,  we  have 

24.r  +  15?/  =  204;  (3) 

24ar+14y  =  200;  (4) 

subtracting  (4)  from  (3),  y=4; 

substituting  this  value  of  y  in  (1),      8x  + 20=68  ; 


whence, 

8j-r=48,  and  ar=6,  Ans. 

2.     Given, 

(5.r  +  2y=y9,                       (1) 
\lx-6y=   9;                       (2) 

multiplying  (1)  by  3, 
adding  (2)  and  (3), 
whence, 

15a;  +  6y  =  57;                     (3) 
22.r=66; 

ar=3,  and  y=2,  Ans. 

3.     Given, 

3x+1y=l9,                      (!) 
ir  +  4y=38;                      (2) 

multiplying  (2)  by  3, 
subtracting  (1)  from  (3), 
whence, 

3;r  +  12y  =  114;                   (3) 
6y=35; 
y=1y  and  a;=10,  Ans. 

4.     Given, 

(  5x-3y=SQ,                      (1) 
^2a:  +  9y  =  96;                      (2) 

multiplying  (1)  by  3, 
adding  (2)  and  (3), 
whence. 

15a;-9y=108;                   (3) 
I72:zr:204; 
a;=12,  and  y=8,  Ans. 

5.     Given, 

f    x-\-My=5i,                      (1) 
J3a:-25y  =  10;                      (2) 

multiplying  (1)  by  3, 
subtracting  (2)  from  (3), 
whence. 

3ar  +  51y  =  162;                    (3) 
76y=152; 

y=:2,  and  a;=20,  Ans. 

(109) 

54 


BIMFLE  EQUATIONS. 


6.     Given, 

6:r-4y=     40,                  (1) 
a:-6y=~97;                 (2) 

multiplying  (2)  by  5, 
subtracting  (3)  from  (1), 
whence. 

6a;-25y=-485;               (3) 
21y=     525; 

y=     25,anda;=28,^w«. 

v.     Given, 

multiplying  (1)  by  3, 
multiplying  (2)  by  4, 
subtracting  (4)  from  (3), 


f8ar4-15y=     9,  (1) 

J6a:-12y=-l;  (2) 

24a;  +  45y=:     27;  (3) 

24ar— 48y=— 4;  (4) 

93y=     31;y=^i=i 


substituting  the  value  of  y  in  (1),    8a;4-5=     9  ;  x 


:t1 


8.     Given, 


j  Yx  +  7y=30, 
]3x  +  4y=l7; 

multiplying  (1)  by  3,  21;r  +  21y=90; 

multiplying  (2)  by  Y,  21a;4-28y  =  119; 

subtracting  (3)  from  (4),  1i/  =  29 

substituting  the  value  of  1y  in  (1),     7^+29  =  30 


9.     Given, 

multiplying  (1)  by  2, 
adding  (2)  and  (3), 
whence, 


10.  Given, 

multiplying  (2)  by  2, 
adding  (1)  and  (3), 
whence, 

11.  Given, 

multiplying  (1)  by  3, 
multiplying  (2)  by  5, 
subtracting  (4)  from  (3), 
whence, 


j  8a;  +  3y=:25, 
|5j;-6y  =  55; 
162:  +  6y  =  50; 
21a;=105, 
ar=5,  and  y: 


Ans, 

(1) 

(2) 
(3) 
W 

Ans. 

(1) 

(2) 
(3) 


—5,  Ans, 


(  I5x-8y=     9,  (1) 

(  10ir  +  4y=-43.  (2) 

20jr  +  8y=  — 86;  (3) 

35x=  —  11; 

ar=  —  2  j,  and  y  =  —  5|,  ^7W. 

j  9a;-5y=     950,  (1) 

(  2a;-3y=— 450;  (2) 

27a:— 15y=      2850;  (3) 

10a:-15y=-2250;  (4) 
I7.r=     5100; 

ar=300,  and  y  =  350,  Ans, 
(109-UO) 


TWO  UNKNOWN   QUANTITIES.  55 

:20,  (1) 


12.     Given, 


-—1- 
2     4" 


^+1=10;  (2) 


multiplying  (2)  by  2,  f+f^^*' '  (^) 

adding  (1)  and  (3),  -^=40  ; 

3 

whence,  2.r=120;  x=QO. 

substituting  in  (1),  30— 1=20  ;     y=40. 


13.     Given, 

multiplying  (1)  by  2, 
multiplying  (2)  by  3, 

subtracting  (4)  from  (3), 

clearing  (5)  of  fractions, 
whence, 

substituting  this  value  of  y  in  (3),       a:+ 10  =  16  ; 

ar=6. 


2  +  3-      ®' 

(1) 

|-F-> 

(2) 

"1=  "; 

(3) 

.-!=-., 

(4) 

14=    »: 

(5) 

10y  +  9y=19xl5; 

y=15; 

14.    Given, 


3^-|=3|,  (1) 

4y-|=.7;  (2) 

multiplying  (1)  by  2,  Qx—yz=i   Y;  (3) 

multiplying  (2)  by  5,  20a;--y=35;  (4) 

subtracting  (3)  from  4,  14x=28; 

whence,  ar=2,  and  y=5,  ^ns. 

(UO) 


56 


SIMPLE   EQUATIONS. 


15.    Given, 


adding  (1)  and  (2), 

clearing  (3)  of  fractions, 
or, 

clearing  (1)  of  fractions, 
subtracting  (6)  from  (7), 
whence, 


+  8y=194, 


^-1-8j;=131; 
8 


+  8(j;-fy)  =  325; 


8 

x-\-y  +  e4(x  +  7j)  —  d25  X  8 

65(.c  +  y)=:325  x8 

x  +  7j^=     5x8 

a;+64y  =  194x8 

63j/=189  X  8 

y=24,  and  x 


(1) 

(2) 

(3) 

(4) 

(5) 
(6) 
(n 

16,  A-v. 


16.    Given, 


X 

3  + 


=21, 


+  3j-  =  29; 


(1) 


(2) 


adding  (l)  and  (2),         i(-r  +  y)  +  3(.r  +  y)  =  50 ;  (3) 

whence,  10(a;  +  y)  =  50  x  3  ;  (4) 

or,  a:4-y=   5  x3;  (5) 

from  (1),  a:  +  9y=:21  x  3 ;  (6) 

subtracting  (5)  from  (6),  8y  =  1 6  x  3  ; 

whence,  y=6,  and  a:  =  9,  Ans, 


17.     Given, 


adding  (1)  and  (2), 

whence, 

or, 

from  (1), 

subtracting  (5)  from  (6), 

whence. 


+  7y=99, 


4-7a;=51; 


4(^+y)  +  7(a;4-y)  =  150; 

50(a;  +  y)  =  150x7; 

x-\-y=     3x7; 

a;+49y=   99x7; 

48y=   96  X  7  ; 

y=14,  and  x-. 

(110-111) 


(1) 

(2) 

(3) 

(4) 
(5) 
(6) 

:7,  Ans, 


18.     Given, 


TWO   UNKNOWN   QUANTITIES.  67 


4     4 

---=1.  (1) 

---=H;  (2) 


2     1 
subtracting  (2)  from  (1),  "=o>  ^°^  y=^ 

y    -^ 

4  . 
substituting  value  of  y  in  (1),  -=2,  and  a;=2 


u4n5. 


19.     Given, 


dividing  (1)  by  147,  i-^=2T'  (^> 

,.,.,>,»  17     17     68 

multiplying  (3)  by  17,  V""7=2T'  ^^^ 

.       ,  .  .        ,  ^  73     41     68     219      ,  , 

subtracting  (4)  from  (2),  y=-g — ^=-^'    w) 

13       1 

dividing  (5)  by  73,  -=— =^,  and  y=7 

17      1^ 
substituting  value  of  y  in  (3),  -=—=:-,  and  ar=3 


20.     Given, 


Reducing  the  first  members  of  equations  (1)  and  (2)  to  mixed 
quantities,  we  have 

ar  +— =a;+— -;  3) 

4  a?  +  y 

27  54  ,^. 

(111) 


d8  SIMPLE  EQUATIONS. 

dropping  x  in  (3),  and  )  14^ 

dividing  by  1 7,         3  4~«Ty'                     ^^ 

dropping  y  in  (4),  and  )  12 

dividing  by  27,        f  h^x—y  *                     ^^^ 

clearing  (5)  of  fractions,  a?+y=16  ;                        (7) 

clearing  (6)  of  fractions,  «— y=10;                        (8) 

adding  (7)  and  (8),  2ar=26,  and  ar=13 

subtracting  (8)  from  (7),  2y=  6,  and  y=:  3 


>■  Ans, 


21.     Given, 


2y— ^     ^^      59  — 2a; 

-^^ =20 

23— a:  2 


ar-^>— =20 —,         (1) 


^+Ss=--^^-.     (^) 


multiplying  (1)  by  2,  2a;— ^—^= 40-59 +  2aj,        (3) 


or. 


4v— 2a; 
clearing  (4)  of  fractions,  437  — 19a;=4y— 2a;,  (5) 


or, 


I7a;  +  4y  =  437;  (6) 

multiplying  (2)  by  3,  3y  +  ?^^=90-73  +  3y,        (7) 


or. 


a:-18 
3y-9 


=  1V;  (8) 


a;— 18 

clearing  (8)  of  fractions,  3y— 9=l7a;— 306 ;  (9) 

transposing,  17a;— 3y=297 ;  (10) 

subtracting  (10)  from  (6),  7y=140,  and  y=20  j     . 

whence,  from  (10),  17a; =3  5  7,  and  a;=21  j 

|6a;'-24y'  +  130 
2.-4y  +  3      -^^  +  ^y+^>  (1) 

9a;y-110  ,  151 -16a; 
-^]^3^+-43^^r-=3-r;  (2) 

clearing  (1)  of )       ^  ,     ^     , 

frac^ons,       \      ^^  "24^'  + 130  =  6.'-24y'  +  lla:+14y+3  ;  (3) 

whence,  lla;+14y=127 ;  (4) 

(111) 


TWO   UNKNOWN    QUANTITIES.  59 

(15l  —  l6x)(Su  —  i) 

whence,  110  — 12^;=^=^ ^^-^^ ^;    (6) 

4y— 1 

or,       440y  — 48a;y— 110  +  12a'  =  453y— 48.ry— 604  +  64.r;  (7) 

or,                 52.r  + 13^  =  498;  (8) 

or,                    4a:  +  y  =  38;  (9) 

adclmg.(4)  and  (9),     15.r  +  15y  =  165 ;  (10) 

^4-y-ll;  (11) 

subtracting  (11)  from  (9),               3a;=27,   and  x=9  \ 

substituting  value  of  x  in  (^,i//  )                         y  =  2  ) 


a;  +  3y_7.c— 21_3a;  — 15_8jr  — 9y 

,      3  6      "      4  12 

23.     Given,  / 

^    2a;  +  y     9.r— 7     3y  +  9     4.r  +  5y 


(1) 

2  8  4  16      '      ^^^ 

dealing  (1)  of  fractions,  4j--f  12y— 14i:4-42  =  9;r— 45  — 8a;  +  9y ,  (3) 

transposing  and  uniting,                  1U-  — 3^  =  87 ;  (4) 
clearing  (2)  of  fractions,  16a:+8y— 182:+14=:12y  +  36  — 4;r— 5// ;  (5) 

whence,                                                  22:  +  y  =  22;  (6) 

multiplying  (6)  by  3,                         6x  +  3i/=GQ\  (7) 
adding  (4)  and  (7),                                  l7a;ii::153,  and  x=:9    ) 
substituting  value  of  a:  in  (6),            18+y=   22,  and 


x=z9    I 
y  =  4    J 


24.     Given, 

ax  +hy  =d^ 
'   a'x  +  b'y=d'', 

(1) 
(2) 

multiplying  (1)  by  b\ 

b'ax  +  b'bi/=b'd; 

(3) 

(2)  by  6, 

ba'x  +  b'by  =  bd'\ 

(4) 

subtracting  (4)  from  (3), 

{b'a-ba')x=b'd-bd' 

(5) 

whence, 

b'd-bd' 
'"'-'b'a-ba'  ' 

multiplying  (1)  by  a', 

a' ax -\- a' by— a' d  ; 

(6) 

(2)  by  a, 

a'ax-\-ab'y=ad' ; 

P) 

subtracting  (7)  from  (6), 

{a'b--ab')y  =  a'd-ad' ', 

(8) 

whence, 

a'd-ad' 
y-a'b-ab'' 
(111-112) 

60 


SIMPLE   EQUATIONS. 


25.     Given, 

multiplying  (2)  by  6, 

subtracting  (3)  from  (1), 

clearing  (4)  of  fractions, 
whence,  from  (1), 


a       0 


ah     ab  ' 


a     a 


(1) 

(2) 
(3) 


|_|=a6-i'=J(a-6);      (4) 

y(a—h)  =ab^(a—b)^  and  y=ab^ 
X  *  \  Ans, 


--\-ab=2ab^ 


and  x=a''b 


26.     Given, 


ax-\-cy: 

ex  4-  ay  -. 

multiplying  (1)  by  a,  a*x-{-acyz 


a*+c* 


ac 


multiplying  (2)  by  c,  c*x-}-aci/= 


ac' 


a'  +  c' 


(1) 
(2) 

(3) 


subtracting  (4)  from  (3),     (a'-c')a:r^^^*-^-ti'=:'^'('^'7^') ; 


whence, 
or, 


o'(a*-c')  _  g' 


substituting  value  of  a;  in  (2),    -  +  ay= ; 

c  ac 


whence, 


a'+c*     a     c*     c 
ay— =_=:_, 


ac        c     ac    a 


on 


c 


(112) 


TWO   OR   MORE    UNKNOWN    QUANTITIES.  61' 

SIMPLE   EQUATIONS 

CONTAINING    MORE    THAN    TWO    UNKNOWN    QUANTITIES. 

(172,  page  116.) 

r  2ar  +  4y— 3^  =  22,  (1) 

1.     Given,  <  4x—2i/  +  5z=l8,  (2) 

(  6x  +  1i/-   z=6S.  (3) 

As  the  coeflScients  of  x  in  (2)  and  (3)  are  multiples  of  the  coeiRcient 
of  X  in  (1),  combine  (1)  with  (2),  and  then  with  (3),  to  eliminate  x. 


Multiplying  (1)  by  2, 

4x  +  S7/—6z=4i, 

(4) 

bringing  down  (2), 

4x-^2y-\-5z=l8; 

by  subtraction, 

10y-llz=26; 

(5) 

multiplying  (1)  by  3, 

6x  +  12y— 92  =  66; 

(6) 

bringing  down  (3), 

6x  +  1y-z=e3; 

by  subtraction, 

5y-8z=  3; 

(7) 

multiplying  (7)  by  2, 

107/-162=  6; 

(8) 

subtracting  (8)  from  (5), 

5z=20,  and  z  = 

=4; 

substituting  value  of  z  in  (7), 

5y=35,  and  y= 

='7; 

"           values  of  z  and  y 

'in(l),                                  xz 

=3. 

(    3ar  +  9y  +  82=41, 

(1) 

2.     Given, 

<     5x  +  4r/—2z=20, 

(2) 

(  lla:  +  7y-62=37; 

(3) 

multiplying  (2)  by  4, 

20x  +  16y—8z=   80; 

W 

bringing  down  (1), 

3ar +  9^4-82=   41; 

by  addition. 

23ar  +  25y=121  ; 

(5) 

multiplying  (2)  by  3, 

16a;4-12y— 62=   60; 

(6) 

bringing  down  (3), 

Ux  +  1y-6z=   37; 

by  subtraction, 

4ar  +  5y=   23; 

0) 

multiplying  (7)  by  5, 

20a;  +  25y=115; 

(8) 

subtracting  (8)  from  (5), 

dx=z  6,  and  xz 

=  2; 

whence,  by  (8), 

2by=1b^  and  y- 

=3; 

and  by  (1), 

82=  8,  and  z- 
(116) 

=  1. 

62  SIMPLE  EQUATIONS. 

ja:  +  y  +  z=32,  (1) 

3.  Given,  i  x  +  y—z=25,  (2) 

ix^y-z=  9;  (3) 

adding  (1)  and  (3),  2ar=40,  and  a:=20  ; 

subtracting  (3)  from  (2),  2y=16,  and  y=  8; 

**  (2)     "      (1),  22=   6,  and  z=  3. 

rar4-y  +  2=26,  (1) 

4.  Given,  "j  ^— y      =4,  (2) 

(  x-z       =  6 ;  (3) 


adding  the  three  equations, 

8ar=36,  and  ar= 

12; 

substituting  value  of  x  in  (2), 

y= 

8; 

U                        U                    U 

(3), 

z= 

6. 

^■ 

-y-z=  6, 

(1) 

5,     Given, 

i3y- 

-x-«=12, 

(2) 

(y^- 

-y— jr=24. 

(3) 

Assume 

ar-fy  +  2=«; 

equation  (1)  becomes 

2j:=6  +  *, 

w 

-        (2)        « 

4y=12+», 

(5) 

"        (3)        « 

8z=24  +  «. 

(6) 

multiplying  (4)  by  4, 

8x=24  +  4«; 

(^) 

(5)  by  2, 

8y=24  +  2«; 

(8) 

bringing  down  (6), 

8y=24+«; 

by  addition, 

8«=72  +  '7«; 

whence. 

«=72; 

substituting  value  of  s  in 

W, 

a:=39; 

((                   u               u 

(5), 

y=21  ; 

tt                        U                   (( 

(6), 

2=12. 

(1) 

6.     Given, 

\3y=M+a:  +  2, 
j  42=tt+u;  +  y, 
(    u=a:-14; 

(2) 
(3) 
(4) 

Assuming  x  +  y-\-z-{-u=s^  and  adding  ar  to  both  sides  of  (1),  y  to 
both  sides  of  (2),  and  z  to  both  sides  of  (3), 

(116) 


TWO  OR   MORE   UNKNOWN   QUANTITIES.  63 


equation  (1)  becomes  3x=s 

«         (2)        "  4y=s 

«  (3)  "  62=5 


7.     Given, 


(5) 

(6) 
(7) 


60a;=20«; 

(9) 

60y=15s; 

(10) 

60z  =  12«; 

(11) 

60w  =  20s— 840; 

(12) 

"         (4)         «  «  =  3-U;  (8) 

multiplying  (5)  by  20, 

"  (6)  by  15, 

(1)  by  12, 

"  (8)  by  60, 

by  addition,  60s  =  67s— 840; 
whence,  *=120; 

substituting  value  of  s  in  (5),  a:=40; 

''       (6),  y  =  30; 

"  "  "       (7),  z=24; 

"  "  "       (8),  tt  =  20. 

ANOTHER  METHOD. 

Subtracting  (2)  from  (1),  2x—St/=y—Xy  or  3x=4y;  (5) 

(2)     "     (3),  42-3y=y-0,  or5«=4y;  (6) 

adding  (3)  and  (4),  4z  =  2x+y—U\  (1) 

multiplying  (7)  by  5,  20z=10x  +  5y  —  10\  (8) 

(6)  by  4,  20x=16y;  (9) 

by  subtraction  10x-lly=70;  (10) 

multiplying  (10)  by  3,  30a:— 33^  =  210;  (11) 

(5)  by  10,  30ar-40y=:0;  (12) 

subtracting  (12)  from  (11),  Yy=210,  and  y=SO ; 

from  (5),  (6),  and  (4),  ar=40,  2=24,  u=26. 


u  +  3x-y-  z=7,  (1) 

2tt  — 2a;-f-y  +  32=8,  (2) 

3u—  ar  +  y  — 4z  =  8,  (3) 

4u+  x^y-2z=1\  (4) 

adding  (1)  and  (2),  3m  +  a; +  2^=15  ;  (5) 

"      (1)    "     (3),  4M  +  2a:  — 52  =  15;  (6) 

"      (2)    "    (4),  6m-  x-{-   z  =  15;  (7) 

"      (5)    "    (7),  9m +  32=30;  (8) 

u      (3)    «    (4),  7m-62  =  15;  (9) 

(116) 


u 


SIMPLE   EQUATIONS. 


adding  (8)  and  (9), 
"      (8)    "    (10), 

substituting  value  of  m  in  (8), 

"  values  of  tt  and  z  in  (5), 

♦*  "      of  w,  z  and  x  in  (2), 


16m-3z=46;  (10) 

26tt='75,  andw=3  ; 


z=l; 
a:=4; 

y=7. 


(  5x-  y  +  7z=61, 
<  4x  +  dy  +  3z  =  8, 
(  Sx—  y—5z  =  S; 

9x—3y— 152=9  ; 
lSx—l2z  =  l1 ', 
2ar  +  12z  =  58; 

15a;  =  75,  and  a;= 
substituting  value  of  x  in  (6),  122=48,  and  z= 

"  values  of  x  and  2  in  (1),  y= 


8.     Given, 

multiplying  (3)  by  3, 
adding  (4)   and  (2), 
subtracting  (3)  from  (1), 
adding  (5)  and  (6), 


5; 

•8. 


9.     Given, 


Adding  the  equations, 

■whence, 

subtracting  (Y)  from  (1), 
(7)     «     (2) 
(7)     "     (3), 
(7)     "      (4), 
(7)     "     (5), 


u  +  v+x+z  +  2y=50, 
u  +  v+y+z+2x=49, 
u+x+y+z+2v=46, 

6»  +  6»+6x  +  6y +  62=240; 
u+v  +  x+y+z=:iO; 
2=12; 
y=10; 
x=  8; 
e=  6; 
u—  4. 


(1) 
(2) 
(3) 

(*) 
(5) 
(6) 


(1) 
(2) 
(3) 
(4) 
(5) 

(6) 


ANOTHER  METHOD. 

Assuming  u  +  v-\-x-\-i/  +  z=s,  and  subtracting  this  from  each  given 
equation, 

equation  (1)  becomes                                   2=52— s;  (6) 

,      "        (2)        "                                         y=50-«;  (7) 

«        (3)        "                                            ar=48— »;  (8) 

«        (4)        "                                           v=46~5;  (9) 

**       (6)       "                                         w=44-s;  (10) 

(116-117) 


TWO   OR   MORE   UNKN^If'  QUANTITIES. ^^'^^        65 


by  addition,  s—240  —  5s',       (11) 

whence,  5~   40;  '    (12) 

substituting  value  of  s  in  (6),  (7),  (8),  (9)  and  (10), 

2=12,  y=10,  x=8,  v  =  6,  u  =  4,  Ans. 

2x  +  i/—2z       =40,  (1) 

4i/—x-{-3z       =35,  (2) 

10.  Given,                            ^           3u  +  t       =13,  (3) 

ij  +  u  +  t        =15,  (4) 

3^— y  +  3<— M  =  49,  (5) 

Eliminate  t  and  u  first  according  to  practical  suggestion  (170,  2. 

Adding  (4)  and  (o),                           3x  +  4t=   64;  (6) 

multiplying  (3)  by  4,                      12m  +  4<=52;  (7) 
subtracting  (7)  from  (6),              3^;— 12w=   12, 

or,                                                        x—4u=     4 ;  (8) 
subtracting  (3)  from  (4),                   y  —  2u=     2, 

or,                                                      2y—4u=     4;  (9) 

subtracting  (9)  from  (8),                    a:— 2y=0,  orir=2y;  (10) 

from  (1)                                     6jr  +  3y— 6^=120;  (11) 

from  (2),                                     8y-2.r  +  63=   70;  (12) 

by  addition,                                    4.r  +  lly=190;  (13) 
from  (10)  and  (13),                       8y  +  lly=190,  or  y=10; 
by  substitution  in  (10),                                                  a;=20  ; 
"     (1),                                                 z=   6; 

"  "  "     (9),  u-  4; 

"  **  "     (3),  tz=   1. 

(    ^+   y-  2=1,  (1) 

11.  Given,                                <  8ar  +  3y  — 6^=1,  (2) 

(  32-4a:-  y=l.  (3) 

Adding  (1)  and  (3),                                2z—Zx—2  ;  (4) 

multiplying  (1)  by  3,                       3a:  +  3y— 3«=3  ;  (5) 

subtracting  (2)  from  (5),                         3z— 6a;=2  ;  (6) 

(4)     "      (6),                           «-2a;=0,orz=2^;  (7) 
substituting  value  of  2  in  (4,)                  4a;— 3a;=2,      ir=2  ; 
whence,                                                                              5?=4 ; 
substituting  values  ofx  and  2  in  (1),                                 y=3. 

(117) 


66 


SIMPLE   EQUATIONS. 


12.     Given, 


2w  +  2x  +  2w+   2  =  -3,  (1) 

Su  +  3x+Sz-h2j/=:     3,  (2) 

4u  +  4i/  +  4z  +  Sx=  —  2,  (3) 

5j;+5y  +  52  +  4M=     2.  (4) 


Assuming  u  +  x  +  i/  +  z=s^  and  adding  z  to  both  sides  of  (1),  y  to 
(2),  x  to  (3),  and  u  to  (4), 

equation  (1)  becomes  2*=— 3+^,  (5) 

(2)  «  35=     3+y,  (6) 

(3)  "  4«=-2+.r,  (7) 
«        (4)        "  55=      2  +  w;  (8) 

by  addition,  145=5.  (9) 

But  (9)   can  be  true  only  when  s=0;    hence  by  substituting  this 

value  of  5  in  (5),  (6),  (7),  and  (8),  we  have 

^=3,  y  =  -3, 
x=2y  u  =  —  2. 


13.     Given, 


Clearing  (1)  of  fractions, 

«        (2) 

«         (3)  " 

multiplying  (4)  by  4, 
subtracting  (5)  from  (7), 
multiplying  (4)  by  TO, 
"  (6)  "     3, 

by  subtraction, 
multiplying  (8)  by  4, 
whence, 


2      3^4 
X     y     z 


X    y  ,  z 

+  -  +  -: 

4     5     6 


:62, 

:47, 
:38. 


6jr  +  4y  +  3z: 

20x-\-\by+\2z 

15^+12y  +  10z 

24ar  +  16y+120 

A:X  +  y 

60x  +  40y4-302 

45j;  +  36y  +  302 

15a; +  4y 

16a;  +  4y 

ar=24,  y 

(117) 


(1) 
(2) 
(3) 


=  744;  (4) 

=  2820;  (5) 

=  2280;  (6) 

=  2976;  (7) 

=  156;  (8) 

=  7440;  (9) 

=  6840;  (10) 

=  600;  (11) 
=  624; 
=  60,  and  2= 120,  ^715. 


TWO   OR   MORE 

UNKNOWN    QUANTITIES. 
/  1      1 

U*r'- 

(1) 

14.     Given, 

Ill 

(2) 

\  -  +  -=3; 
\y    z 

(3) 

adding  the  three  equations, 

H^"^  ■ 

(4) 

dividing  by  2, 

y,f- 

(5) 

subtracting  (1)  from  (5), 

1=2,  or  2=4 ; 

"           (2)     "      (5), 

l=l,ory=l; 

"           (3)     "      (5), 

-=1,  or  x=l. 

lx-\-a=y  +  z, 

(1) 

15.     Given, 

K+a=-ix+2z, 

(2) 

{z  +  a=2x  +  3y\ 

(3) 

by  transposition, 

X — y — z^=  — a  ; 

(4) 

~1x+y  —  iz=—a; 

(5) 

—3x—3y  +  z=—a; 

(6) 

adding  (4)  and  (6), 

—2x—4y=—2a; 

0) 

or, 

-x-2y=-a; 

(8) 

multiplying  (4)  by  2, 

2x—2y—2z=—2a; 

(9) 

Bubtnicting  (5)  from  (9), 

ix—Sy=—a; 

(10) 

multiplying  (8)  by  4, 

—  ix—8y=—4a  ; 

(11) 

adding  (10)  and  (11), 

lly=     5a; 

whence. 

y=A«.  «=t't«.  2=Vt« 

,  Ans. 

16.  Assume  x-{-y-\-z=s]  then  the  equations  become, 

si-z  =  2(b  +  c),  (1) 

s  +  y=2{a  +  c\  (2) 

s-\-x=2(a-\-b).  (3) 

(117) 


68  SIMPLE   EQUATIONS. 

Add  i  ng  and  uniting,  4* = 4  (a  4-  6  +  c) ; 

sz=a-\-b-\-c  ; 
by  substitution  in  (3),  x=za-\-b—c   ) 

"    (2),  y=a  +  c-&   [jns. 


(1),  z=b-\-c-a 


(    U-3i/=a,  (1) 

lY.     Given,  <  5t/—llx=a,  (2) 

(    9y—5z=a;  (3) 

multiplying  (1)  by  5,  35.r— 15y=:5a;  (4) 

(2)  by  3,  — 33jr+15y  =  3a;  (5) 

adding  (4)  and  (5),  '  2x=Sa] 

whence,  ar=4a,  y=9a,  and  2  =  16a. 


18.     Given, 


^  +  y_^_y  (I) 

a^b-b     a'  ^^^ 


Clearing  (1)  of  fractions,  bx-\-ay=:ax—bf/] 

or,  (a-6)j;-(a  +  %  =  0;  (3) 

multiplying  (2)  by  (a+b),    (a  +  b)x  +  {a  +  b)7/=-—^ ;  (4) 

11.       /«x        ,  /.N                                      «          ^a'ft    '     .          2a6 
adding  (3)  and  (4),  2ax= -^  and  x= r  ; 

substituting  value  of  ar  in  (3),     2a6  — (a  +  Z>)y  =  0,        and  y=: 7. 

a  -}-  0 


19.     Given, 


Clearing  (3)  of  fractions, 
and  transposing, 


ax  +  by  +  cz=ab-\-ac-{-bc,  (l) 

a'x  +  bhj  +  c^z=3abc,  (2) 

l-^+.v=f.+  y=2+!r«=o.  (3) 

be  uc  ac  ab  ^ 

I  ax  +  2b)/  +  cz=2ac-}-ab  +  bc ;  (4) 


ac 


subtracting  (1)    from  (4),  by^=ac^  and  y=^-r 

substituting  value  of  y  in  (1),  ax-\-cz=iab-\-bc\  (5) 

«  "        "      "(2),  a'x4-cV-2a6c;  (6) 

(117-118) 


TWO   OR   MORE   UNKNOWN   QUANTITIES.  69 

multiplying  (5)  by  r,  acx+c^z=ahc-i-bc''  ;  (7) 

be 
subtracting  (7)  from  (6),  a{a—c)x=bc(a—c)y  and  x=  —  ] 

substituting  value  of  x  in  (5),        bc  +  cz=abi-  bc^  and  z  =  — . 

c 

{    cx  +  y  +    az~2a,  (1) 

20     Given,  J   c*x  +  y  +  a^z  =  2ac,  (2) 

(  acx—y  -f  ac2=a'  +  c'.  (3) 

Adding  (1)  and  (3),  car  +  «c.r  +  (/2  +  ac2  =  2a  +  rt'  +  c* ;  (4) 

or,  c(l +a)jr  +  o.(l+c)2=2a  +  a'  +  c';  (5) 

adding  (2)  and  (3),  c{c-\-a)x-\-a{c-\-a)z=c'^ +  2ac-{-a^^ 

or,  dividing  by  (c  + a),  cx  +  az:=c  +  a\  (6) 

multiplying (6)  by  (1+c),  c{\-\-c)x-{-a(\-\-c)z=c  +  a  +  c' +  ac\  (7) 

subtracting  (7)  from  (5),     c{a—c)x=.a'^ —ac ■\-a—c=::{a'-c){a  + 1) ;  (8) 
whence,                                                           x=a-{-\  ; 

substituting  in  (6),  i 

"  (1),  y=^a-c 


a    ' 


r  a'x  +  ay  +a2  =«,  (1) 

21.     Given,  I   ax-^a''y  +  az  =a\  (2) 

(   ax -{-ay  +a'«=a'.  (3) 

Dividing  each  equation  by  a,  aar  +  y  +  r  =  1  ;  (4) 

x-\-ay-{-z=a]  (5) 

ir  +  y  +  a2  =  a';  (6) 
subtracting  (5)  from  (4),              (a  —  l)x  +  {l—a)y  =  l—a^ 

ora-— y=  — 1;  (7) 
(6)     -     (2),             (a-l).r4-(a'-l)y==0; 

orx+{a-hl)y=0]  (8) 

^         "  (7)     «     (8),  (a  +  2)y=l;  or  y:       ^ 


substituting  in  (7), 


a  +  2 

0  +  2  a  +  2 


whence,  .=  (^, 

'  a  +  2 

(U8) 


70 


SIMPLE   EQUATIONS. 


PROBLEMS 

PRODUCING   EQUATIONS   CONTAINING    TWO    OR   MORE    UNKNOWN 
QUANTITIES. 

(174,  page  119.) 

1.  Let  a:=  the  first  number,  and  3/=  the  second;  then  by  the 
conditions  of  the  problem,  we  have 

2jr  +  3y=105,  (1) 

3i:  +  2y=   95;  (2) 

whence,  by  elimination,  ar=15,  and  y=25,  Ans. 

By  adding  the  equations,  and  taking  |  of  their  sum  from  the  first 
equation,  we  have  the  value  of  y ;  taking  the  same  from  (2),  we 
have  X, 


2.  Let  x=z  the  first,  y=  the  second,  2=  the  third. 


By  the  conditions, 


clearing  (1)  of  fractions, 
subtracting  (3)  from  (4), 
clearing  (2)  of  fractions, 
subtracting  (3)  from  (5), 
whence, 


:,  +  r:__  =  120,  (1) 

y+'-i^=  90,  (2) 

«  +  ar  +  y  =  190;  (3) 

22;  +  y  +  «=240;  (4) 

xz=z   50; 
5i/+x-hz  =  450;  (5) 

4yr=260; 
y=^65,  and  2=75. 


3.  Let  x=z  A's  share,  y=  B's,  and  z=  C*s. 


By  the  conditions, 


x-^(y  +  z)  =  l20=a,  (1) 

y-^(^  +  ^)  =  120=a,  (2) 

z-^{i/  +  x)  =  l20=a;  (3) 
(119) 


TWO   OR   MORE   UNKNOWN   QUANTITIES.  71 

clearing  of  fractions,  1x—4:y—4z=^a,  (4) 

Sx-{-8i/  —  3z  =  8a,  (5) 

^2x—2y-[-9z=9a',  (6) 

multiplying  (4)  by  (2),  l4x  —  8]/-8z—Ua;  (1) 

adding  (7)  and  (5),  Ux—llz  =  22a,orx—z=2a'^  (8) 

multiplying  (6)  by  2,  — 4j~4y  +  182  =  18a  ;  (9) 

subtracting  (4)  from  (9),  ^llx-^22z=zlla^OT  —x  +  2z=a]  (10) 
adding  (10)  and  (8),                                       2=:3a=360, 
whence,                                                           a;=:600,  and  y=480. 

4.     Let  x=  A's  daily  wages,  y=  B's,  and  z=:  C's. 


I    6(x  +  y)=40, 

x  +  y=6l; 

(1) 

By  the  conditions,         <    9  (a;  + «) = 54, 

x+z=6; 

(2) 

(  15(y  +  2)  =  80, 

y+2=5i-. 

(3) 

One  half  the  sum  of  the  equations  is 

x+y+z=9; 

(*) 

subtracting  (3)  from  (4), 

^=3|; 

(2)     "     (4), 

y=3; 

(1)     "      (4), 

z=2i. 

5.  Let  X,  y,  z  and  u  represent  their  ages  respectively. 

ix  +  y  +  z=18,  (1) 

a;  +  y-fu=16,  (2) 

x  +  z  -\-u  =  14^  (3) 

y  +  «4-w  =  12.  (4) 

One  third  the  sum  of  the  equations  is,  a;  +  y  +  «  +  M  =  20;         (5) 
subtracting  (4)  from  (5),  ar=   8  ; 

"    .      (3)      "      "  y=   6; 

«  (2)      "      "  z=  4; 

it  (1)      «      «  u=  2. 

6.  Let  x=:  A's  shillings,  y=  B's,  and  0=  C's ;   then  by  the  con- 
ditions of  the  problem,  after  the  first  game  each  will  have  as  follows : 

x—y—z—  A's  shilHngs, 
2y=  B's        " 
2z=C's        " 
(119) 


72  SIMPLE   EQUATIONS. 

After  the  secofnd  game, 

2x— 2y  — 22=  A's  shillings, 
2i/  —  {x—y—z)  —  2z  =  3t/—x—z=B's        " 

4z=  C's 
After  the  third  game, 


4ar~4y— 42=16, 

(1) 

6y-2i:-2z  =  16, 

(2) 

72_ar-y  =  l6; 

(3) 

Adding  the  equations, 

x  +  i/  +  z  =  4S] 

(4) 

"       (3)  and  (4), 

8z=64,  and  z=   8  ; 

"       iof(l)to(4), 

2a:=52,  and  x=26  ; 

substituting  in  (4), 

y=u. 

T.  The  same  equations  as  in  Ex.  6th,  page  116. 

8.  Let  ar=  the  value  of  the  better  horse,  and  y=  the  value  of 
the  poorer.    ^ 

-fMo  =:/r;rv?-7-<-)/7^,_/^-^,^^-U+15  =  -(y  +  10),  (1) 

By  the  conditions,  } 

^^:t:;:^u  («+io=i5(y+i5);       (2) 

Jy^^  i,i-^~\i   =  ^-^  15 

adding  5  to  both  sides  of  (2),    a:  +  15  =  --(y  +  15)  +  5  ;  (3) 

lo 

co.npan„g^(l)  and  (3),  J  4^^^,„)^15(^^^,)^,  .  ^^^ 

clearing  of  fractions,         52y  +  520=45y  +  675  +  195 ;  (5) 

whence,  7y=350,  and  y=50,  poorer  horse  ; 

and  by  substitution  in  (1),  a; =65,  better  horse, 

9.  Let  ar=  the  price  of  a  dozen  of  sherry, 

y=       "         "        "       "     brandy.     Put  a=r78. 

By  the  conditions,  j  f  ^o^^f^    o  ^1 

^  '  (  7a:  +  2y  =  9a  +  9;  (2) 

multiplying  (1)  by  2,  4:c+2y  =  6a ;  (3) 

subtracting  (3)  from  (2),  3jr=3a4-9, 

or,  .  ar=a  +  3  =  81,  sherry ; 

whence,  from  (1),  y=a— 6  =  72,  brandy, 

jz^  -  >   -/,  (119-120) 

J  V   ^-  ^  4  J 


TWO   OR   MORE    UNKNOWN    QUANTITIES.  7Sl 

10.  Let  x=  the  time  in  which  A  can  do  the  work, 
-.__     «        ti        *'      B        "       "      " 

then  -=  the  part  of  the  work  A  can  do  in  one  day, 

u  u  u  J5        «  «  u 

y 

In  four  days,  both  working  together  would  do  \  of  the  work. 

Therefore. 
JL  ^  ~l  ^~L  1       36     3  12      1 

whence,        /  r"  ,  y  =  48,  and  ;r=24,  ^«*. 

11.  Let  a;=  numerator,  and  y=  denominator. 

2i:       2 


"^-f^  /4     4     1 


By  conditions. 

y  +  7-3' 

x^2     3 

2y  ~5 

(1) 

;          (2) 

clearing  of  fractions,  and  transposing,        %x—2y=z 

bx—Qyz=i- 
multiplying  (3)  by  3,                                18^— 6y  = 
subtracting  (4)  from  (5),                                   13j;= 
whence,                                                                {c= 
Hence  the  fraction  is 

14,             (3) 
-10;          (4) 
42;           (5) 
62, 

4,  y=b. 
1,  Ans, 

12.  Let  x=A'8  money,  and 

y=B'8, 

1 

By  the  conditions,                    \ 

^  +  |^=240=a, 
^x 

(1) 

{ 

y  +— =240=:a; 

(2) 

clearing  of  fractions, 

subtracting  (3)  from  (4), 
substituting  in  (3), 

3x  +  2y  =  3a, 
3a;  +  4y=:4a  ; 
2y=a,    y= 
3i:  +  a  =  3a,  x= 
(120) 

(3) 
(4) 

=  120} 

=  160, 

74  SIMPLE    EQUATIONS. 

13.  Let  rr=  the   number  of  persons,  and   y=.  what  each  paid  ; 
then  xy=  the  amount  of  the  bill. 


By  conditions, 

C(^  +  4)(y-l)  =  a-y; 

(1) 

(2) 

expanding  (1), 

XyJ^\y—X  —  \=iXy, 

(3) 

"         (2), 

xy  —  Zy^x—Z—xy\ 

(*) 

dropping  xy 

4y— a;— 4  =  0, 

(5) 

-3y  +  ar-3r=:0; 

(6) 

adding  (5)  and 

(6), 

y-7  =  0; 

whence, 

y=i7,  a:  =  24.  Arts, 

14.  Let  x~  the  digit  in  the  place  of  tens,  and  y=  the  digit  in 
the  place  of  units  ;  then  10i:  +  y  will  represent  the  number. 

■KT       1       .  ,.  .  (  10x  +  y  =  4x4-4y,  (1) 

Now  by  t.ie  conditions,  ^  _  ^  ,  ^      /^( 

^  '  (  10a:  +  y  +  27=a:+10y;(2) 

transposing  and  reducing  (1),  2x—y\            (3) 

(2),  y-^=3;           (4) 

adding  (3)  and  (4),  ar==3,  whence,  y—^. 

Hence,  36,  Ans, 

15.  Let  X  represent  the  number  of  hundreds,  y  the  number  of 
tens,  and  z  the  number  of  units  ;  then  lOO^  +  lOy  +  z  will  represent 
the  number. 

r  x  +  y+2-11,                     (1) 

By  the  conditions,      \  z=2x,                        (2) 

(  100xi-l0y  +  z-\-29l=x  +  l0y  +  100z;  (3) 

transposing  and  re-  )  _ q  .                       {a\ 

ducing  (3),  3 

substituting   value   )  9  _    _o  . 

of  z  in  (4),  J 

whence,  x=Sj  z  =  Gj  and  y=2. 

Hence,  326,  Ans. 


16.  Let  Xj  y,  and  z  represent  the  parts. 

(x-\-y-\-z=90,  (1) 

By  the  conditions,  <    2a;  +  40  =  3y  +  20,  (2) 

(    40-|-lO=:2a;  +  4O;  (3) 

(120-121) 


TWO   OR   MORE    UNKNOWN    QUANTITIES.  75 

from(2>  y= — ^ — ; 

2ar  +  30 
"    (3)  ^=—4—' 

,     .      .      .    ,,v               2ar+20     2i?  +  30     ^ 
by  substitution  m  (1),      x-\ f-  — =90, 

ar=35,  first    part, 
y=30,   second  " 
2=25,  third     ** 

ANOTHER  METHOD. 

By  the  conditions,  twice  the  first  part  plus  40,  three  times  the 
second  part  plus  20,  and  four  times  the  third  part  plus  10,  are  all 
equal  to  the  same  number,  which  we  may  represent  by  x ;  then 

a:— 40 

— - —  z=z  the  first     part, 

^—20         ,,  ,     „ 

=     "  secofi'l    *' 

3 

and  ^j^^     „    ^j^.^^       ,, 

4 

^       ^  a:— 40     x—20     x—\0     ^^  .. 

Therefore,  -y- +  — ^  + -^  =  90,  (1) 

whenccy  a;=110. 

— - —  =  35, =  30,  and  — - —  =  25,  Ans. 


17.  Let  x=  the  part  at  5  per  cent,  and  y  the  part  at  4  per  cent. 
Put  $100,000=0,  4640=6. 

I  ar  +  y=a,  (1) 

By  the  conditions,  -j   5a;       ^V  _j^,  /n\ 

( ioo'*"ioo~  '  ^  ' 

from  (1),  y=za-~x\ 

5x      4ff 4aj 

"      (2),  by  substitution,  i5o"^"Tocr=*' 

whence,  a;=1006  — 4a=$64000  )  ^^^ 

and  y=       a—  x=$36000  J 

(121) 


i6 


SIMPLE   EQUATIONS. 


18.  To  avoid  the  labor  of  using  large  numbers  in  the  operation, 
put  a  =  5000;  then  2a  =  10000, 3a =16000, 


10 


lOr/ 
1500,  aud  ^--  =  800. 


Let  x=:  the  principal  of  the  first  person,  and  y=  the  rate. 


ar+2a= 
and      x-\-3a= 


By  conditions, 


clearing  of  fractions  ) 
and  dropping  ary,   j 
multiplying  (3)  by  2, 
subtracting  (4)  from  (5), 
substituting  in  (3), 

Hence, 


"    second     "         y+l=    " 

"      third      "        y  +  2=    " 

^y        ry4-2ay4-a:  +  2a     16a 

iob~  100  Too* 

ary      a-y  +  3a  t/  +  2.r  +  6a       3« 

iob~  Too       ~~""  10 

0  =  2</y4-   ar  +  2a  — 16a 


0  =  3ay4-2a;  +  6a— 30a; 
0  =  4ay-f  2jr  +  4a  — 32a  ; 
at/  —  4a^  and  y=4  ; 
2:= 6a =$30000. 


(1) 

(3) 
(4) 
(5) 


Principals,  $30,000,  $40,000,  $45,000 ; 


Ratt 


4, 


5, 


6,  per  cent* 


19.  Let  T,  y  and  ar  represent  their  respective  ages. 


By  the  conditions, 

adding  (1)  and  (3), 
substituting  in  (2), 
(3), 
subtracting  twice  (5)  from  (4), 
whence, 


5y  +  2z— ar=147, 
x-\-y-\-z=   96; 

2a:=    96,  and  x 
6y  +  22=195 
y  +  Z=    48 
3y=    99 
y=   33,  2;=15. 


48; 


0) 

(2) 

(3) 

(4) 
(5) 


20.  Let  x=  what  A  is  worth,  y=  what  B  is  worth,  and  z=  what 
C  is  wortb  ;  also  let  s—  what  they  all  are  worth.     Put  100=a. 


ix-\-3y  +  dz=4la,     Y  ^<>^ 

(1) 

By  the  conditions. 

K  y+4a;  +  4z=58a,      y^<^ 
(  2  +  5a;  +  5y  =  63a;    ^S^^ 

(2) 

(3) 

adding  2x  to  (1), 

ds=  47a +  22;  : 

(4) 

"      3y  to  (2), 

48=   58aH-3y; 

(S) 

"      42  to  (3), 

5s=  63a +  45?; 
(121-122) 

(6) 

4y  ^/2  y  ^/Z  c^  -  / 

/'  r 

e^-t? 

(jj  ZOyL  -^2J>^    ^  4  SL  ^  -2-  y 

-2  ^ 

?^ 

1^0 

TWO   OR   MORE   UNKNOWN    QUANTITIES.  77 


multiplying  (4)  by  6, 

18s=282a  +  12.r;             (1) 

(5)  by  4, 

16s  =  232a  +  r2y;             (8^ 

(6)  by  3, 

i5s=zl89a-\-12z',              (9) 

by  addition, 

49s  =  703rt  +  12s; 

whence. 

s=   19a; 

substituting  value  of  s 

in  (4), 

2xz=   10a,  x=5a  =  500  ; 

U                          U                     t( 

(5), 

3y=    18a,  y=:6a  =  600; 

tt                        tt                   u 

(6), 

4z=   32a,  2  =  8a  =  800. 

21.  Let  x=z  the  cost  of  a  pound  of  tea,  t/—  the  cost  of  a  pound 

of  coffee;  then  oOx=  the  whole  cost  of  the  tea,  and  30?/=  the  whole 

cost  of  the  coftee.     Ilis  gain  in  selling  the  tea  is  y'^  of  its  cost,  or 

5x ;  his  gain  in  selling  the  coffee  is  \  of  its  cost,  or  6y. 

^>  -^  ^Oy-J.4^jy^  (    5i:+   6y=   2.90,  (1) 

Bf  the  conditions,  1  .-   T  „«       «k  .  J  /«( 

^^j     -i^y-  U5a;+36.y  =  27.40;  (2) 

subtracting  6  times  (1)  from  (2),  25x=  10.00  ; 

whence,  ar=$.40,  and  y  =  $.15,  Ans. 

22.  Let       a*,  y,  ar,  w  and  v  respectively  represent  their  money. 


Then  by  the  conditions, 


substituting  value  of  v  in  (4), 
**  "  ti  in  (3), 

**  "  z  in  (2), 


z+|=30, 

(1) 

M  =  30. 

(2) 

|-i  =  -. 

(3) 

Su     V 

{*) 

5v    „^    ^ 

-—=30,  whence  v= 
6 

=  36; 

(5) 

?^=24,       "         «  = 

=32; 

1=2.        ..       .= 

=  33; 

1=19,       "       ,-. 

=38; 

. 

19=30,       "       a:= 

=  11. 

122) 


78  SIMPLE  EQUATIONS. 

23.    Let    x^  y  and  z   represent  respectively  their  money.    Put 
a=1000. 


/    ar  +  |=2000= 

=2«, 

(1) 

J5y  the  conditions. 

/  y  +  ^  =  2000  = 
) 

=  2a, 

(2) 

(    2r  4-^  =  2000  = 

=  2a; 

(3) 

i  2x+y=.   4a, 

(4) 

clearing  of  fractions, 

^  3y  +  2=   6ff, 

(5) 

. 

(42+ar=   8a; 

(6) 

multiplying  (5)  by  4, 

12y  +  42=24a; 

(7) 

subtracting  (6)  from  (7), 

\2y  —  x=\Qa\ 

(8) 

"         (8)  from  12  times  (4),         25a:=32a; 

32a 
whence,  ar=— — ; 

Zo 

or,  a:=1280; 

by  substitution,  y=1440; 

and  z=1680. 


24.  Let  ar=  the  hourly  rate  of  the  first  courier,  and  y=  the  hourly 
rate  of  the  second  courier.     Now  the  distance  divided  by  the  rate 
will  be  the  time ;  hence, 
Z  _  -/   -    hi  -  J^  147      147 

^    .L/.  £p  ^      y 

These  equations  are  the  same  as  those  in  Ex.  19,  page  111,  and  may 
be  solved  in  a  similar  manner. 

y  ^  3 


-7 


25.  Let  x=.  the  greater,  and  y=  the  less. 
By  conditions. 


X     y 

2+1=13,  (1) 


(122)     ^    ^2^^-/3 

J      7 


TWO   OR   MORE    UNKNOWN    QUANTITIES.  79 

3^ 


multiplying  (1)  by  3, 

y+y=39; 

(3) 

(2)  by  2, 

2-e 

(*) 

by  addition, 

13'     o« 
6   =««' 

■whence, 

ir=:18,  and  y  =  12. 

,  Ans. 

26.  Let  X—  the  first, 

y=1 

the  second,  and  2=  third. 

( 

^  +  My+«)  =  51=flr, 

(1) 

By  conditions, 

y  +  i(a?+2)  =  51=a. 

(2) 

( 

z^\{x^y)  =  h\=a', 

(3) 

( 

a^  +  (^+y  +  e)  =2a. 

(4) 

clearing  of  fractions, 

2y+  (a;  +  y  +  2r)=3a. 

(5) 

( 

3z  +  (a:  +  y  +  2)  =4a. 

(6) 

Let  (ar  +  y4-«)=s,  anc 

I  multiply  (4)  by  6,  (5)  by  3,  and 

(6)  by  : 

equation  (4)  becomes 

6ar  +  6s  =  12a; 

(7) 

"          (5)        " 

6y  +  35=  9a; 

(8) 

(6)       " 

6z+2s=   8a; 

(9) 

by  addition, 

65+ll«=29a,  or«=:87. 

substituting  value  of  s  in 

'W. 

ir=2a  — «,  or  ar=15; 

((                          U                    (( 

(5). 

2y=3a— 5,  or  ^=33 ; 

ti                       H                 M 

(6), 

3r=4a  — 5,  or  2  =  39. 

2*7.  Let  a;=  A's,  y=  B's,  and  z=  C's  sheep. 

rar  +  8-4=y  +  2-8,  (1) 

3y  the  conditions,  •<    i(y  +  8)=a;  +  2  — 8,  (2) 

(  M^  +  8)=^+y-8;  (3) 

clearing  of  fractions  and  uniting,      a; — y  -  « =  —  1 2,  (4) 

—  2a:  +  y  — 20=— 24,  (5) 

—  3a;— 3y  +  2-— —  32;  (6) 
adding  (4)  and  (6),  _2a;-4y=-44;  (7) 
or  a;  +  2y  =  22;  (8) 
subtracting  twice  (4)  from  (6),  — 4a;  +  3y=0;  (9) 
adding  four  times  (8)  to  (9),                     lly=:88  ; 

vhence,  y—   8,  a:=   6,  and  2=10. 

(122-123) 


80  SIMPLE   EQUATIONS. 


28.  Let  -  represent  the  fractions. 


By  the  conditions, 


clearing  of  fractions,  <fec., 

whence,  by  subtraction, 
consequently  the  fraction  is 


29.  Let  -  represent  the  fraction. 


/x  +  l_ 

)   ^ 

1 

"3' 

(1) 

I      X 

1 
~4^ 

(2) 

3z—y= 
4x^y= 

X- 

=  -3; 

=    1; 

=      4, 

and  y- 

=  15; 

(3) 
(4) 

Ans, 

ar4-2      5 


By  tho  conditions. 


y       7 

X         1 


(1) 
(2) 


y+2    3' 

clearing  of  fractions,  <kc.,  Yj:— 5y=:  — 14;  (3) 

Zx-y=        2  ;  (4) 

subtracting  (3)  from  five  times  (4),     8i:=     24,  whence  ar=3  ;  (5) 
substituting  in  (4),                                                               y=^^\ 
hence  the  fraction  is,                                  ^,  Am. 

30.  Let  .r,  y,  z,  and  u  represent  the  parts  of  an  acre  which  the 
men  respectively  can  mow  in  one  hour. 

iar  +  3y+    22+    2w  =  l,  (1) 

3a:  +  2y+    42  +  11m  =  2,  (2) 

5a;  +  4y  +  12z+    5m  =  3,  (3) 

9-r  +  7y+   6z+    8if  =  4,  (4) 

subtracting  twice  (I)  from  (2),  x—  4y-\-    1u=z     0;  (6) 

"         sixtimes(l)from(3),  — a;— 14y—   1u  =  —3;  (6) 

adding  (5)  and  (6),  18y=3,  whence  y=l ; 

subtracting  (4)  from  three  )  2m--1  •  ^7^ 

times  (1),  \  -6x  +  2y-2u-     1,  (7) 

substituting  the   value  of  )  ar— 1  +  7?^=     0;  (8) 

y  in  (5)  and  (7),  f  -6ar  +  i-27^  =  -l  ;  (9) 

(123) 


TWO   OR    MORE   UNKNOWN    QUANTITIES.  81 

transposing,  and  multiply- )  ,     ^ 

ing(8)by6,  3  ^       \     ) 

adding  (10)  and  (9),  40u  =  ^^  w=tV» 

by  substitution,  ^=h  ^^^  ^— ti* 

Hence,   A  can  mow  an  acre  in  5  hours;  B  in  6  hours;  C  in  12 
hours ;  and  D  in  15  hours. 


(1) 

(2) 


31.  Let  x=  A's  money,  and  i/=Ws. 

By  the  conditions,  ]  """f  "!^/ "^ ^} 

^  *  (a:  +  5  =  3(y-5); 

V     5 
by  subtraction,  10  =  3y— 15— -— -;       (3) 

Z      2 

whence,  20==5y— 35,  and  .y=:ll  ; 

by  substitution,  a:=:13. 

32.  Let  x=  the  number  of  bushels  of  wheat  flour, 
and  y=   "         "         "         "  barley     " 
Then       l0x  +  4i/=  the  cost  of  the  whole, 

and       ll^  +  lly=  the  value  of  the  sale  at  11  shillings  per  bushel. 

Now,  Tn  —  Inh  '  *"^  ^^  *^^®  conditions, 

fii(10x  +  4y)  =  ll.r  +  lly,  (1) 

or,  6'750ar+2300y=:4400^  +  4400y,  (2) 

or,  1350.c  =  2100y, 

or,  9j:  =  14y. 

Converting  the  equation  into  a  proportion,  by  placing  the  factors 
of  one  member  for  the  extremes,  and  the  factors  of  the  other  for 
the  means,  we  have 

ir  :  y  : :  14  :  9 
or,  wheat  :  barley  : :  14  :  9,  Ans, 

33.  Let  X  represent  the  number  of  tens,  and  y  the  number  of 
units;  then  the  number  expressed  in  units,  must  be  10^ +  y. 


By  the  conditions. 


(123-124) 


82  SIMPLE   EQUATIONS. 

Reducing  (1),  we  have  y=l» 

(2),    "      "  x=4. 

The  number,  therefore,  is  41,  Ans. 


34.  Let  x=z  the  seniors,  y=  the  juniors,  z=  the  sophomores,  and 
u=  the  freshmen.     Place  119=a. 


Now  by  the  conditions  of  the  problem,  we  have 


*+!=-.    (1) 

y+l^a, 

(2) 

^  +  ^=«,          (3) 

u+l=a; 

(*) 

subtracting  (2)  from  twice  (1), 

2«-|=«; 

(6) 

adding  (3)  to  three  times  (5), 

6j!+|=4o; 

(«) 

subtracting  (4)  from  four  times  (6), 

24j;-|=15a; 
5 

or, 

119z=75o; 

whence,                     a;=75,  y=88,  z= 

=03,  and  u=104. 

35.  Let  Xy  y,  «,  and  u  represent  the  numbers  respectively. 

/  3^  +  y=:a=359,  (1) 

By  the  conditions,  )  ^^  "*"  "^  ='''  (^) 

J  52  +  «— a,  (8) 

\6u+x—a]  (4) 

subtracting  (2)  from  four  times  (1),    l2x—z  =  Sa ;  (6) 

adding  (3)  to  five  times  (5),  60x+u  —  l6a;  (6) 

subtracting  (4)  from  six  times  (6),         359.e=  95a ; 
whence, 

a:=95,  y=:V4,  2=63,  and  w=44, 

(124) 


GENERAL   PROBLEMS.  83 


GENERAL  SOLUTION  OF  PROBLEMS. 
(179,  page  126.) 
1.  Let  xz=.  the  greater  part,  and  y—  the  less, 
conditions,  \  , 


(1) 
(2) 


transposing  (2),  a:— y=6—a;  (3) 

n  +  6 — a 
adding  (I)  and  (3),  2a;=«  +  6— cr,      ar=- ; 

subtracting  (3)  from  (1),       2y=n  +  «— o,      y— ■ 


2 
84  4-58-16 
2.  Substituting  the  numerals, 


2  =^^' 


84  +  16-58     „, 


3.  Let  a-,  y,  and  «  represent 

the  numbers. 

/ar  +  y+2=5,                                  (1) 

By  the  conditions, 

^               :r=:y-«,                            (2) 

(           y-^-&,                    (3) 

subtracting  (3)  from  (2), 

a;-2y  +  «=6-a;                          (4) 

"          (4)     «     (1), 

.               s-Va  —  h 
3y_s  +  a     6,  ory=: — ^ ; 

pubstituting  in  (2), 

s-\-a  —  b                     s—2a—h 

X-       g           a,  or  ^_        ^ 

«{3) 

54-0  — &     ,              s  +  a4-26 
2-       ^       +6,  org-                . 

4.  Let  ar=  my  indebtedness  to  A ; 

then  nxzz,        «  "  B ; 

and  m.r=        «  *<  C. 

Now  by  the  conditions,  x-\-nx-^mxr=^a\ 

whence,  ^—r-. — ; — ^  Ans, 

'  1  +  w  +  m ' 

(128) 


84  SIMPLE   EQUATIONS. 

6.  Substituting  the  numerals,         x=- — - — -=$131,  Ans, 

1  +24-3 


6.  Let  xz=  the  number  of  days  he  was  idle,  and 

a—x=         "  "  "         worked; 

then  (a—x)b=:  the  money  due  for  his  labor,  and 

cx=     "       "       he  forfeited. 
Now  by  the  conditions,  we  have 

(a—z)b—cx=d ; 

whence,  x=  ~ ,  Ans. 

o-\-c 

ANOTHER    METHOD. 

6.  Let  x=  the  number  of  days  he  was  idle.  He  would  have  re- 
ceived ab  cents,  had  he  worked  all  the  time.  Every  day  he  was 
idle  he  lost  his  wages,  6,  and  forfeited  c  cents ;  his  whole  loss,  there- 
fore, for  the  idle  days,  was  (6  +  c)x.     ITierefore, 

ab—{b-i-e)xz=d; 
whence,  ^~~h '  '^^^' 


v.  Let  ar=  the  value  of  the  saddle,   and  nx=  the  value  of  th« 
horse ;  then 

(»  +  l)x=a; 

whence,  x= -,  and  nx= -,  Ans. 

n+1  »+l 


8.  Let  x=  the  rent  last  year;  then  by  the  conditions, 

nx 
^  +  100=''  = 

100a 
whence,  '=100+;i' ^"'• 

(126) 


GENERAL    PROBLEMS.  85^ 

9.  Let  X  represent  his  income  ;  then  by  the  conditions, 

X  X 

Ux=2](a-^b), 
whence,  x= — ^— — -,  Aru, 

10.  Let  X  represent  his  income  ;  then  by  conditions, 

X  X 

—  +  a-\-~-hb=x; 
m  n 

whence,  nx-[-nix-\-nin{a-\-h=mnx)^ 

x(mn—m—n)=mn(a-{-b)^ 

mn — m — n 

11.  Let  ar=  the  number  of  days  required  when  all  work  together. 
Since  A  can  do  the  work  in  a  days,  in  one  day  he  will  do  a  part  of 

the  whole  work  expressed  by  -,  and  in  x  days  he  will  do  a  part  of 
the  work  expressed  by  -.     Reasoning  similarly,  in  x  days  B  will  do 

X 

a  part  of  the  work  expressed  by  y,  and  C  a  part  of  the  work  ex- 
pressed  by  -.     Therefore, 

XXX 

-+t4— =1; 

a     b     c 

whence,  x{ab  +  ac  +  be) = a5c, 

abc  . 

x=— —   Ans. 

ab-{-ac  +  bc 

6x8x12        5^6 

12.  Substituting  the  numerals,  x=-— — -— — —-=——=2*^,  Arts. 

^  *         48  +  72  +  96     216        '' 

13.  Let  X  represent  the  number  ;  then  by  the  conditions, 

ax—c^bx-{-d  ; 

whence,  (a—b)x=c  +  dj 

c-hd       . 

x= r»  ^'**» 

a—b 

(126-127) 


86  '  SIMPLE   EQUATIONS. 

14.  Let      x=  the  number  of  bushels  of  oats, 
y=  "  "  "        .  peas. 

Then  by  the  conditions,  i  ,  ^  )Jk 

^  \ax-hby=cd\  (2) 

subtracting  b  times  (l)from  (2),  (a — b)x=c{d—b) ;  (3) 

c(d-b) 

whence,  x=z-^ r-^,oats; 

a  —  0 

subtracting  (2)  from  a  times  (l),            (a—h)t/=c(a  —  d)  ; 
whence,  y  =  -^ ^ ,  peas 


15.  Let  x=  the  number  of  nuts.  The  party  of  a  boys  lost  bm 
nuts  by  the  snatching,  and  the  party  of  b  boys  lost  am  nuts.  The 
party  of  a  boys  had  x—bni  nuts  U)  ha  divided,  and  each  would  have 

.     The  party  of  b  boys  had  x — ani  nuts  to  be  divided,  and 

each  would  have — i — .     Therefore, 

0 

X — bm     X — atn 
T'""~b~' 
whence,  (6 — a)xz=.m{b'* — a'). 


mib'-a")        ,       ,,     , 

x=-^. '=m{a-\-b\  Ans. 

b—a  ^         ' 


16.  Let  a;,  y,  sr,  and  u  respectively  represent  the  numbers. 

ax-^yz=zm,,  (1) 

■r.      ,  ,.  .  J  by-\-z=.m^  (2) 

By  the  conditions,  /  \  [ 

^  cz  ■\-uz=m,  (3) 

du-{-xz=zm.  (4) 

Subtracting  (2)  from  b  times  (1),      abx—z=im(b — 1)  ;  (5) 

adding  (3)  to  c  times  (5),  abcx-i-u=m(bc~c  +  l) ;  (6) 

subtracting  (4) from  c?  times  (6),  abcdx—x=m(bcd—cd  +  d—l);    (1) 

,  m(bcd — cd  +  d—1) 

whence,  x——^ -— ^; 

abed— I 

(127) 


GENERAL    PROBLEMS.  8?- 


substituting  in  (4),  du=m ^^ will ' 

mi  abed — bed  -\-cd — d) 

''"^ ^cirri • 

_m(abc-hc  +  c—\)  _ 

...           ,         -      .     /  X                             m(abc  —  bc-\-c  —  \) 
substituting  value  oiu  in  (3),  cz=m ^ j~- — , 

miabcd—abc  ^-bc—c) 
or,  cz= —^ ^  • 


abed  —  1 


m(abd — ab  +  b  —  1) 
dmdingbye,  ^= ^^— ^ ; 


substituting  in  (2), 


m(acd—ad-\-a — 1) 
abed — 1 


17.  Let     a;=  the  number  of  pupils  ; 
then       ax — c=  the  whole  attendance  in  one  school, 
and        bx—d=:    "         "  "  the  other  school. 

na=  the  number  of  days'  attendance  for  which  A  pays, 
mb=  "  "  "  "  B     " 

Now  either  patron  will  pay  such  a  part  of  the  whole  money  raised 
in  his  school,  as  his  number  of  days'  attendance  is  part  of  the  whole 
attendance.     Therefore,  by  the  conditions, 

mb  na 

bx—d     ax—c ' 
whence,  abmx — cbm=^anbx — adn\ 

or,  x{ab)  (w — n) = bcm  —  adn ; 

bcm  —  adn 


ab(m — w) 


Ans, 


18.  Let  the  several  parts  be   represented   by  a?,  ax,  a'ar,  and  a'x 
respectively.     Then  by  addition, 

a*x-{-a*x  +  ax-^xz=ni  ; 

whence,  x  =  -r =- — — - ,  Ans, 

(127-128) 


0) 

(2) 


88^.  SIMPLE   EQUATIONS. 

19.  Let  X  and  y  represent  the  numbers. 
By  the  conditions,  i        ^~^| 

by  addition,  2j:=:5  +  rf,  and  x— ; 

2 

by  subtraction,  2y=«— rf,  and  y= . 

20.  Let  a-,  y  and  ar  represent  the  numbers. 

rar  +  yzra,  (1) 

By  the  conditions,  <  ar  +  «=6,  (2) 

(y+2=c;  (3) 

subtracting  (3)  from  the  sum  of  (1)  and  (2),      1x-=ia  4-  6— c, 

a4-6— c 

(2)         "         «         (l)and(3),     2y=a  +  c-6, 


y= 


2 


"  (1)         "         "         (2)  and  (3),      2z=6  +  c-a, 

6-f-c— a 

21,  Let        «=  the  number  of  tens  ; 
and  y=        "         "        units; 

then        10x-f-y=  the  number  of  units  in  the  given  number. 

By  the  conditions,  ]  J^^+2^      ^t'/"''*  ^}1 

^  (  10x  +  y-l-c  =  10y  +  ar;  (2) 

UlO-a)i:  +  (l-a)y  =  0,  (3) 

(  9a;-9y=-c;  (4) 

multiplying  (3)  by  9,  9(10— a)a;4- 9(1 —a)y=0  ;  (5) 

"  (4)by(l-a),     9(l-a)ar-9(l-a)y=ac-c;  (6) 

by  addition,  9(11  — 2a)ar=c(a— 1), 

whence,  '=9(n-2a)' *"°°' 

substituting  in  (4),  -^j — — ^+c=9y, 

c(lO-a) 

2^=9(Trr^)'""^''- 

(128) 


NEGATIVE   RESULTS.  89 

INTERPRETATION   OF   NEGATIVE  RESULTS. 
(  18^5  page  133.) 

1,  Let  X  represent  the  number;  then  by  the  conditions, 

X       X 

-rir  ''■' 

whence,  x=144y  Ans, 

In  an  arithmetical  sense,  the  fourth  part  of  a  number  can  never 
exceed  its  third  part;  and  the  absurdity  of  this  supposition  is  indi- 
cated by  the  negative  result.  Hence  the  question  must  bo  modified 
so  as  to  read, 

What  number  is  that  whose  third  part  exceeds  its  fourth  part  by 
12? 

2.  Let  X  equal  the  number  of  years ;  then  by  the  conditions, 

30  +  jr=      3(15 +  x); 
ar=  — VJ,  Ans. 
As  the  man  was  30  years  old,  and  his  wife  15  years  old,  at  the 
time  of  marriage,  she  is  already  one  half  as  old  as  he ;   he  can, 
therefore,  never  become  three  times  as  old  as  she.      Ilencc,  the 
problem  must  be  modified  as  follows : 

A  man  was  30  ijeai's  old  when  he  married,  and  his  wife  15.  flow 
many  years  be/ore  their  marriaye  was  his  aye  three  times  the  aye  of 
his  wife  ? 


)x  -\-  u — s 
\  whence, 
x—y—d, 


If  we  make  «=120  and  rf=160,  we  have 

120  +  160       ,^^ 
xz= =   140,  greater; 

120-160         ^^  , 
y— =  —  20,  less. 


s-\-d 

s-d 
"~2~ 


Algebraically  considered,  —20  added  to  140  is  120,  the  sum  ;  and 
—20  subtracted  from  140  is  160,  the  difference, 

(133) 


90  SIMPLE   EQUATIONS. 

4.  Let  x=  B's  money ;  then  3^:=  A's  money.  Now  by  the  con- 
ditions of  the  problem,  we  have 

32:  +  400  =  2(j;  +  150); 
whence,  ar=:  — 100, 

3ar=  — 300. 

The  negative  results  show  indebtedness^  instead  of  capital,  at  the 
commencement  of  business.  Hence,  the  question  should  read  as 
follows : 

A  owes  three  times  as  much  as  B.  A  gains  by  trading  $400,  and 
B  $150,  when  A  has  twice  as  much  money  as  B.  What  was  the 
indebtedness  of  each  at  first  ? 


5.  Let  a:=  the  father's  wages,  and  y=  the  son's. 

(2) 


T>    .1  :,•  .  S  Vjr  +  3y=22,  (1) 

By  the  conditions,  i  ^  ,  „  ^  ' 

subtracting  (l)  from  three  times  (2),  8j:=   32  ; 

or,  «=     4; 

whence,  y=  — 2. 

The  negative  value  of  y  shows  that  the  boy  was  charged  for  board 
each  day  two  shillings  more  than  his  wages. 


6.  Let  ar,  y  and  z  represent  the  wages  of  the  man,  his  wife,  and 
son,  respectively. 

I  102:+    8y+    62=10.30,  (1) 

By  the  conditions,  ^  12jr  +  10y+   42=J3.20,  (2) 

(  15j;  +  10y  + 12^=13. 85;  (3) 

subtracting  (3)  from  twice  (1),  5x-\-Qy=   6.75  ;  (4) 

"  (3)  from  three  times  (2),    21ar  + 20y  =  25.75  ;  (5) 

multiplying  (4)  by  10,  50a: +  60y  =  67.50 ;  (6) 

(5)  by  3,  632; +  60y  =  77.25 ;  (7) 

subtracting  (6)  from  (7),  13a;=:   9.75  ; 

whence,  a;=.75,  y=   .50,  and  z=— .20. 

The  negative  value  of  z  shows  that  the  boy  was  charged  for  board 
$.20  each  day  above  his  daily  wages. 

(133-134) 


NEGATIVE   RESULTS.  91 

V.  Let  ar,  y  and  z  represent  the  wages  of  each  respectively. 

i  10.r  +  4y  +  3z=rll.50,  (1) 

By  the  conditions,  <     9a:  +  8y  +  6z=:12.00,  (2) 

(    7a:  +  6y  +  42=:   9.00;  (3) 

subtracting  (2)  from  twice  (1),  ll.r  =  11.00  ; 

whence,  a:=:1.00,  y=0,  and  2=  .50. 

The  value  of  y  shows  that  the  wife's  board  and  wages  were  equal. 


8.  Let  x=  the  numerator,  and  y=  the  denominator. 

ar+l      3 
By  the  conditions, 


From  (1) 
"     (2) 


\ 

y 

'5' 

y 

5 

(y+i~ 

''1' 

bx 

-^y= 

:-5; 

1x- 

-5y  = 

:     5; 

25x- 

.I5y= 

:-25, 

2\x- 

■I5y  = 

:        15, 

4x= 

:-40, 

(1) 

(2) 


ar=  — 10,  y=— 15;  hence, — —~yAns, 

— 15 

The  modified  example  will  be  as  follows  : 

What  fraction  it  that  which  becomes  |    when  1  is  subtracted  from 

its  numerator^  and  ^  when  1  is  subtracted  from  its  denominator^ 

The  equations  will  now  be, 

ar— 1_3         X     _5 

whence,  ar=10,  y=15  ;  hence  {^y  Ans, 

9.  Let   Xy  y,  z  and  u  represent  respectively  the  net  capital  or 
insolvency  of  each. 

Ix  +  y  +  z  +  u  =  5l80,  (1) 

x  +  y  +  z        =7950,  (2) 

y +  2  4-^  =  2220,  (3) 

X        +z  +  u  =  l320,  (4) 

(134) 


92;  SIMPLE   EQUATIONS. 

Subtractiog  (3)  from  (1),  x=     3560,  A's  net  capital. 

"           (4)     "     (1),  y=  — 1540,  B's  net  insolvency, 

«           (2)     "     (1),  M  =  — 2170,  D's  net  insolvency, 

by  substitution,  z=     5930,  C's  net  capital. 

Tlie  positive  values  of  z  and  z  show  net  capital  for  A  and  C ;  the 
negative  values  of  y  and  u  show  net  insolvencj/  for  B  and  D. 

10.  Let  xz=  the  number  of  hours  Jiftcr  six  o'clock,  when  A  passed 
B.  Now,  at  six  o'clock,  B's  distance  from  Boston  was  n  -}-  46  miles. 
Hence,  at  the  moment  of  passing  we  have 

m  —  ajr=A's  distance  from  Boston; 
n  +  ib-bxz=Ws       "  "  "     . 

Therefore,  in — ax=n  +  4b  —  bx\ 

m — n — 46 


a-b 


hours,  yins. 


11.  Substituting  the  given  vahus  for  7n,  n,  a,  and  6,  we  hav 


36  —  28—12      —4 

x= = =  —  2. 

5  —  3  2 

That  is,  A  passed  B  —  2  hours  after  six,  or   2  hours  be/ore  six, 
which  is  4  o'clock. 


12.  Let  ar=r  the  greater,  and  x—a=:  the  less;  then  by  the  con- 
ditions, 

J  x-{-5{x—'a)  =  b', 

64-5a      . 
whence,  x= — - — ,  the  greater  ; 

o 

6  — 3a      ,     , 

and  X — a  = ,  the  less. 

8 

If  a=:24  and  6=48,  then 

48  +  120  48— -K©-       ^ 


21,  greater;  x^a= =— Syless. 


8  '°  '  8 


Arithmetically  speaking,  there  is  no  number  —9.  But  considering 
the  quantities,  21  and  —  H^  algebraically^  they  fulfil  the  conditions 
of  the  problem.  3 

(134-136) 


DISCUSSIONS.  93 

(193,   page  142.) 

1.  Let  x^  y,  z,  and  u  represent  the  parts  of  the  contents  of  the 
cistern  which  will  flow  through  the  pipes  respectively  in  one  hour. 


I5x  +  l5y  +  15z  +  15u  =  l, 

(1) 

5x+   8y+   1z+    3«  =  J, 

(2) 

Bv  the  conditions, 

1                                 ^                                                   * 

J                             ' 

3x+    4y+    30+      u^\, 

^^) 

4.r+    2y+    32-f    2^  =  1. 

(^) 

Subtracting  J  of  (1)  from 

(2) 

2a:  +  5y  +  45f=:-i-'^, 

(5) 

"          •i'Tof(l)     " 

(3), 

2j:  +  3y  +  23  =  y\, 

(6) 

"          ftof(l)     " 

(4), 

2-1^          +   z=i^\ 

(^) 

multiplying  (5)  by  3, 

C.r  +  45y  +  122— y^; 

(8) 

(6)  by  5, 

10ar  +  15y+10;2r:=ff ; 

(0) 

subtracting  (8)  from  (9), 

\x        -2z=H; 

(10) 

multiplying  (7)  by  2, 

4x         +20-n; 

(11) 

adding  (10)  and  (11), 

8^=      ^f,  and:r=      /^=r      -,\ 

subtracting  (10)  and  (11) 

4z=-if,  and  z=—^^^-^- 

substituting  in  (5) 

5y=     f^andy::^     ^V=     tj 

(1) 

) 

^5w=-?f,  and  t^  =  -/^=--V. 

Therefore,  the  contents  of  the  cistern  will  flow  through  the  first 
pipe  in  12  hours,  through  the  second  in  15  hours,  through  the  third 
in  20  hours,  and  through  the  fourth  in  30  liours.  The  positive 
values  of  x  and  y  indicate  receiving  pipes  ;  the  negative  values  of  z 
and  u  indicate  discharging  pipes. 


2.  Let  m  represent  what  A  will  have  in  2  days ;  then  m4-2  will 
be  what  B  will  have  in  4  days. 

Let  ar=  the  number  of  days  hence,  when  A  and  B  have  the  same 
money;  then  A  will  have  m  +  5(.r— 2)  dollars,  and  B  will  have 
m  +  2  +  3  (z~ 4)  dollars.     Hence, 

w  +  5(jr— 2)  =  m  +  2  +  3(:r  — 4)  ; 
(5— 3)ar=0; 
0 


5  —  3" 

That  is,  they  now  have  the  same  sum. 

(142-143) 


0. 


94  INEQUALITIES. 

3.     Let  X  represent  the  period  ;  then  by  the  conditions, 
3a;-10=:3(4j-4-8); 
-whence,  12a;— 40  =  12x4-24  ; 

(12-12)a:=64; 

64         64 

=— -  =  oo. 


12-12      0 

The  value  of  x  is  an  expression  for  infinity,  according  to  (188,  1.). 
The  period  of  the  comet,  therefore,  is  a  number  of  years  greater 
than  any  assignable  number. 

4.  Let  X  represent  the  monthly  wages  ;  then  by  the  conditions, 
2(9j:  — 450)  =  3(6.c  — 300)  ; 
whence,  1 8a; — 900  =  1 8  j- — 900 ; 

(18-18)j:=900-900, 

_900-900_0 
^~  18—18  ~0' 
The  value  of  ar  is  a  symbol  of  indetermination,  according  to 
(I885  4.)  The  monthly  wages  of  each  may  therefore  be  any 
number  of  dollars.  If  they  receive  more  than  $50  a  month,  they 
•will  each  lay  up  the  same  sum.  If  they  receive  less  than  $50,  they 
•will  become  equally  indebted. 


INEQUALITIES. 
(301,  Page  149.) 

1.  6jr>  5^  +  14; 

clearing  of  fractions,  lOx  >  3a--(-28  ; 

dropping  3a-,  7a;  >  28  ;   whence,  a;  >  4,  Ans, 

2a;     2a;       2a; 

2x  2a; 

dropping  y,  ""T  ^  ""^  ' 

2x  X 

changing  signs  by  (  199,  III.),  y  <  2,  or  -  <  1  ; 

clearing  of  fractions,  ^  <  3,  Ans, 

(143-149) 


INEQUALITIES.  95 


3.  ^  +  7<-^+.n> 


5x     5   ^11  ,1x 
8      4^6       12 

clearing  of  fractions,  75ar+150  <  220  + ^Oar; 

whence,  6;c  <  70  ; 

X  <  14,  Ans, 

Sx     x  —  l  20a;  +  13 

whence,  3x—2x-^2  <  24a  — 20j;— 13  ; 

or,  —3x  <  — 15  ; 

changing  all  the  signs  by  (109,  HI.))  3a;  >  15  ; 

ar  >  5,  Ans. 


6. 

transposing. 


dividing  by  (a—c). 


6. 

multiplying  by  aft, 
transposing, 
dividing  by  (a +  6), 


7.  (a—x)(m—x) — a(m—c)<^x* ; 

1                                          <                         ,     aV 
removing  parentheses,     am—ax—mx-{-x^—am-{-ac<^x^ ; 


ax—b  > 

cx  +  c?; 

ax— ex  >  64-c?; 

^> 

6+fl?      . 

's  Ans, 

a—c 

^< 

^-b 

aar— a'  < 

ab—bx ; 

aa;-|-6a;  < 

a'i-ab] 

^< 

0,  ^ns. 

;    .    ,     aV 

transposing, 

a'c 

-ax—mx  <  —ac ; 

m 

changing  all  the  signs. 

a'c 

aa;+ma;  >  acH ; 

m 

multiplying  both  sides  by  w, 

m(a  4-  m)a;  >  ac(a  +  w) ; 

dividing  by  m(a  +  m). 

(149) 

a;>  — ,  Ans. 
m 

^6  INEQUALITIES. 

(dOd,   page  150.) 

1.  Given,  i2.  +  4y>30,  (1) 

(3x  +  2y  =  31;  (2) 

dividing  (1)  by  2,  a-  +  2y  >  15 ;  (3) 

subtracting  (3)  from  (2),  ( 1  »9,  IT.),  2.r  <  16  ; 

whence,  a:  <     8. 

If  we  substitute  8  for  ar  in  (2),  the  first  member  will  be  greater 
than  the  second  member  ;  thus, 

24  +  2y  >  31; 
transposing, 
whence, 

2.  Given, 

multiplying  (1)  by  2, 

subtracting  (3)  ffom  (2),(199,  II.), 

whence. 

Substituting  2  for  y  in  (2)  will  make  the  first  member  less  than  the 
second ;  hence, 

8x  +  4  <  46  ; 
transposing,  8j:  <[  42  ; 

whence,  x  <  b\. 


3.  Given, 

multiplying  (2)  by  2, 
adding  (1)  and  (3), 
whence, 
substituting  in  (2), 

whence, 


2y>    V; 

y>H' 

4x-3y  <  15, 

(1) 

8jr  +  2y  =  46; 

(2) 

Sx-Qy  <  30; 

(3) 

8y  >  16  ; 

y>   2, 

4.     Given,  \ 

From  equation  (2), 

(150) 


7ar~10y  <  59, 

(1) 

4x+  5y  =  68; 

(2) 

8x-fl0y  =136; 

(3) 

Ibx  <  195; 

x<    13; 

52  +  5y>    68; 

5y>    16; 

y>    31. 

+  3y>  121, 

(1) 

+  4y=  168; 

(2) 

1x 

y=42--; 

(3) 

POWERS   OF   POLYNOMIALS. 


97 


substituting  this  value  of  y  in  (1),  we  have 
6^+126  — 


>121; 


multiplying  by  4,  20x  +  504— 21a:  >  484  ; 

transposing  and  uniting,  — -^  >  —20^ 

whence  by  ( 1 99,  III.),  ^  <       20. 

Substituting  20  for  x  in  (2),  we  have 

1404-4y  >  168; 
transposing,  4y  >     28 ; 

y>     '?. 


W 


a;— 4     y  — 10 


5.     Given, 


8 


6 


>1. 


3ar— 24     x—y 


(1) 


(2) 


multiplying  (1)  by  24, 

Sj;_12-4y  +  40  >  24; 

(3) 

or. 

3^-4y>  -4; 

(4) 

multiplying  (2)  by  4, 

3j;_24+2ar-2y  =  52; 

(5) 

or, 

5a;-2y  =  '76; 

(6) 

multiplying  (6)  by  2, 

10jr-4y  =  152; 

V) 

subtracting  (4)  from  (7), 

Va;<  156; 

whence. 

X  <  22^, 

substituting  in  (6), 

lll|-2y>     76, 

transposing, 

-2y  >  -35f 

or, 

2y  <       35^ 

y  <     iH- 

INVOLUTION. 

(316, 

page  157.) 

1.  By  simple  multiplication  according  to  the  rule,  we  shall  have 

(2a;'4-3y)'=4a;*+    6jr'y  +  6a:'y  +  9y* 
=  4i;'  + 1 2x'y  +  9y*,  Ans. 

(150-157) 


98  INVOLTUION. 

2.  Multiplying  the  factor  bx—y*  by  itself,  we  get 
(5a;'  -y')'  =  25j:'  -  lO^-y'  +y*. 
Multiplying  this  result  by  5x—i/*, 

(5a;— y')'=  125ar»— 50a:y  +    5x/  — 25a;y  +  10a;/— y* 
=  125a;'— 75a;V  +  15a-y*— y',  ^n«. 

8.  {l-\-2x  —  3xy  =  l-^4x-6x'+    4j;'— 12a;'  +  9a;* 

=  1  +  4a;— 2ar'  —  1 2x'  +    9a;*,  Ans, 

4.  (3a  +  26  +  c)'  =  9a'  + 1 2a6  +  6ac  +  46'  +  46c  +  c\ 

Multiplying  this  result  by  3a  +  26  +  c,  we  have 
(3a  +  26-fc)' 

=  27a'  +  36a'6  +  18tt'c-l-  12ff6'  +  12a6c+  3ac' 

+  18a'6  +  (86')  +  24a6'  +  12a6c+(86'c)  +  26c' 

+    9a'c  +  (46V)  +  1 2«6c  +  6ac'  +  46c'  +  c' 

=  2'7a'  +  64a'6  +  2Ya'c-|-36a6'  +  36a6c+  86*     +9ac'  +  126'c 
+   66c* +  c',  Ans. 

In  Examples  6th  and  6th  we  ^lave  the  simple  binomial ;  see  (70). 

7.  (a'c-'  +  a-'cy  z=a*c-*  +  2a'c°  +  a-*c* 

=a*c-*  +  2-ha-*c\  Ans. 

8.  (a'  4- 1  +a-')' =a*  +  2a'  +  2a»  + 1  +  2a-'  +  a-* 

= a*  4- 2a' +  3  4- 2a-' +  a-*. 
Multiplying  again  by  a'  + 1+  a~', 

(a'  +  1+  a-')'  =  a'  +  2a*  4-  3a'  4-  2a''  4-  a-* 

4-    a*4-2a'4-3     4-2a-»4-   cr* 

4-  a'  4-  2a"  4-  3a-'  4-  2a-*  4-  a-' 

(a'  4- 1  4-  a--y=a''  4-  3a*  4-  6a'  4-  V     4-  Qar*  +  3a-*  4-a~*,  Ans. 
For  Example  9th,  see  (71). 

(217,  page  159.) 

All  examples  in  this  article  are  readily  solved  by  a  strict  applica- 
tion of  the  rule,  which  is  an  important  one.  In  the  first  three 
examples  the  symmetry  of  the  answers  should  be  noticed. 

(158-159) 


SQUARE   ROOT   OF   POLYNOMIALS.  99^ 

6.  (l—a  +  a'— a')«=l--2a-f  2a'— 2a' 

4-   a'  — 2a' 4- 2a* 

+   a*  — 2a*  +  a' 


=  1  —  2a  +  3a'  —  4a'  -f  3a*  —  2a'  +  a\  Arts, 

1.  (3aa;  +  2a'  — 4a;'  — 5)' 

:9aV  +  12a'a:— 24aa;'  — 30a2;  +  4a*— 16a'j;'— 20a'  +  16j:*  +  40a;'  +  25 
;12a'x— 24ax'—  30aar  +  4a*  — Ya'jr'  — 20a'  +  16a:*  4-40a:'  +  25,  Ans. 

8.  (1— 2a:— y'  +  ary— a;')' 
:  1  —  4a: — 2y' +  2a:y  —  2a;' 

+  4a;'  4-  4a-y'  —  4a:'y  +  4a:'  +  y*  —  2a'y '  +  2a:'y' 

+  a:'y'  — 2a:'y  +  a;* 

:  1  —  4a: — 2y'  +  2a:y  -f  2a;'  +  4a;y'  —  4a:'y  +  4a;'  +  y*  —  2a;y'  +  3a:'y' 

— -2a;'y  +  a;*,  Ans, 


EVOLUTION. 
(aaO,  page  166.) 

1.  a'  +  2a6  +  2ac  +  6'  +  26c  +  c\a  +  6  +  c,  Ans, 

a' 


2d+b       2a64-2ac  +  6' 
2a6  +6' 


2a+264-c  2ac         +26c+c' 

2ac         +26c4-c' 

2.  a*  — 6a'64-4a'  +  96'  —  126  +  4(a'  — 36  +  2, -4w*. 


2a'  — 36         —  6a'6+  96' 

-6a'6+  96' 


2a«— 664-2  4a'  -1264-4 

4a'  -1264-4. 

(159-166) 


IPP  EVOLUTION. 

3.  «'  +  4j:*  +  2x«  — 2a:'  +  6x*-.2a:+l(df'  +  2a^— ar+l,  Ans. 


4x'-\-4x* 


2x*-h4x*—x         —2x*  —  2x*-\-5z* 

—  2x*-4x'-\-   X* 


2ar'  +  4jr'  — 2jr+l  2x'-\-4x*-~2x-\-l 

2x'  +  4a;'  — 2ar  +  l. 

4.  The  two  first  terms  of  the  root  will  be   l—rt,  by  inspection; 
hence, 

.   I  — 2a  +  3a*  — 4a* +  3a*  — 2a* +  a\\—a  +  a*'-a\  Ans, 
1  — 2a+    a* 

2  — 2a  +  a'  2a'  — 4a' +  3a* 

2a'  — 2a'+    a* 


2— 2a  +  2a'— a'  —2a"  4- 2a*  — 2a' -fa' 

—  2a' +  2a*  — 2a' -fa* 

5.  The  square  root  of   the   first  term   is   2a'6,   and    —  12a'6'-?- 
4a'6=— 3a6;  hence  we  proceed  as  follows  : 

|2a'6— 3a64-2a6',  Ans, 

4a*b*  —  1 2a'6'  -f  8a'6'  +  9a'6'  - 1 2a'6'  -f  4a'6* 
4a*6'-12a'6'4-  9a'6' 


4a'6' — 6ab  +  2a6'  8a'6'  -^1 2a'6'  +  4a'6* 

8a'6'     ■         —  12a'6'  +  4a'6* 

6.  |3x'  — 5a?*y— 4a:y'  +  6y',  Ans, 

9ar'— 30yy4-      a:*y'  +  76x'y'  — 44a;'y*  — 48ary' +  36/ 
.      9x'--30x'y  +  25x*y' 

6j?"— 10ar"y-4ry'  -242:*y'  +  76x'y'-44ar'y* 

-24x*y'  +  40.r'y»4-16a:Y  " 


6a;'  — lOar'y— 8jry"  +  6y'  362r'y'  — 60Jry  —  48a:y»4-  S6y* 

S6xy-Q0xy-48xy*-{-d6y* 
(166) 


SQUARE  ROOT  OF  POLYNOMIALS.  101 

|a'  — 36c4-2c</— c?',  Ans, 
7.    a*-6o'6c+4aW-2aV  +  96V-126c»tf-|-66cc?'+4cV'-4cc^"+rf* 
a*-6a'6c  +9/>V 


2a*-^6bc+2cd       4aW-2aV  —Ubc'd 

Ad'cd  —Ubc'd  +4cV 


2a*-6bc+4cd-d*        -2aV  +66crf'  ^4cd'+.d' 

-2aV  +66ccf'  _4crf'+c?* 

.       „      3a«6«      aJ^      6'      ,  ,     ab     b'      . 

a*—a*b-{-    — - 
4 


2         4       16 


2ar*  — 6a;  +  x-'  2x' —  62?-*  +  a;"* 

2x'  — 6^-*+a;~* 


10.  a"*-'  — 10a6-'  +  27  — 10a-'6+a-'6'(a6-'  — 5+a-'6,  ^n«. 

a«6-»_10a6-»  +  25 


2a6-*  — 10  4-a''6  +   2  — 10a-'6+a-'6' 

2-10a-'6  +  a-«6« 
In  this  example  we  have  the  last  multiplication, 
2ab-'  xa-'6=2a°6»=2. 


11.  a***  4-  6a*"»c''  +  11  a""c'"  +  6a V  +  c^^^a^^  +  3a'"c'*  +  c*",  ^n*. 

a*** 


20'-"  +  Sa^'cT  6a'"*c'*  + 1  la""c^ 

6o""c"+   9a»V 


2a2V  +  6aV+c*'' 
(168) 


102  EVOLUTION. 


SQUARE  ROOT  OF  NUMBERS. 
(934,  page  169.) 

1.  "72,25(85,  Ans.  2.  10,82,41(329,  Ans, 

a>-  64  9 

2a+6-165      825  2<i+h-.62        182 

2a6+6>-  825  124 


2a+26  +  c-649  5841 

5841 


3.      65,12,49(807,  ^ns.  4.      9*7,41,69(987,  ^n*. 

_64 81 

1607      11249  188    1641 

11249  1504 


1967   13769 
13769 


5.   5,09,85,64(2258,  ^»*.        6.  66,34.10,25(81.45,  ^»W. 
4  64 

161    234 
161 


42 

109 

84 

445 

2585 

2225 

4508 

36064 

36064 

7. 

18,12,88,60,8 

16 

82 

212 

164 

845 

4888 

4225 

8507 

66360 

59549 

8514^ 

J    681184 

681184 

1624    7310 
6496 


16285    81425 
81425 


8.  .33,98,89(.583,  ^7W. 
25 


108    898 
864 


1163    3489 
3489 


(169-170) 


SQUARE   ROOT  OF   NUMBERS.  103 


9.  .00,52,41,76(.0724,  Ans. 
49 
142      341 

10. 

41 

4,77(21.8403+,  Ans. 
4 

77 

284 

41 

1444     5776 

428 

3600 

5776 

3424 

4364 

17600 
17464 

436803    1360000 

1310409 

i 

11. 

63 

11.09(3.33016+,  Ans, 

9 

209 

189 

663 

2000 
1989 

■> 

66601 

110000 
66601 

666026 

4339900 
3996156 

12.  The  square  root  of  a  fraction  is  the  square  root  of  the  numer' 
ator  divided  by  the  square  root  of  the  denominator.     Hence, 

13,69(37  1,18,81(109  ^    ^^^ 

9  1  109'     "*' 

67         469 
469 


13.  1,02,03,02,01(10101  1  . 

ioioi'^'*"- 


209    1881 
1881 

1,02,03,02,01(10101 

1 

201 

0203 
201 

20201 

20201 
20201 

(170) 

104 

EVOLUTION. 

14. 

245   49          TT     ^   . 
720  =  144-        Hence,  -,^n,. 

16. 

43 

=  5.57,14,28,57 +  (2.3604  +  ,  Arts. 
4 

157 
129 

466 

2814 
2796 

27204 

182857 
168816 

CONTRACTED   METHOD. 


(«355  page  171.) 


1.      56.00,00,00,0(7.4833147  + 
49 


144 

700 

576 

1488 

12400 

11904 

14963 

49600 

44889 

14966 

4711 

4490 

1497 

221 

150 

150 

71 

60 

15 

11 

11 

2. 
67 

14.00,00,00 
9 

500 
469 

(3, 

7416574 

744 

3100 
2976 

7481 

12400 
7481 

• 

7482 

4919 
4489 

748 

430 

75 

374 

~56 

53 

t 

3 
3 

( 170-171 ) 


SQUARE   ROOT   OF   NUMBERS. 


105 


844 


848 


85 


6. 


142 


1444 


14483 


14486 


1449 


145 


14 


18.00,00(4.2426  + 

4. 

19.00,00,00(4.3588984- 

16 

16 

200 

83 

300 

164 

865 

249 

[         3600 

5100 

33Y6 

870( 

4325 

J           224 

i         77500 

169 

69664 

55 

87l€ 

;            7836 

61 

6973 

872 

863 

785 

87 

78 
12 

52.46,30,00 

r 

1(7.2431346  4-        6. 

7.00,00,00,00(2.64575131 4- 

49 

4 

346 

46 

300 

284 

524 

276 

6230 

2400 

5776 

5285 

2096 

45400 

30400 

43449 

52907 

26425 

1951 

397500 

1449 

62914 

370349 

602 

27151 

435 

26457 

67 

5291 

694 

58 

529 

"9 

529 

165 

8 

159 

53 

6 
5 

(172) 


106  EVOLUTION. 

1.  6*=(5y=(125)^ 

1,25.00,00,0(11.18034  + 
I 

21  25 

21 


221  400 

211 


2228  18900 

18824 


2236  76 

67 

22  9 

9 


(239,  page  175.) 

27a'  +  108'i'  +  144a  +  64(3a+4,  Arts. 


|27a'  108a' +  144a +  64 

9a  +  4     36a  +  16|27a'+36a  +  16  108a'  +  144a  +  64 


In  this  example  we  have, 

Trial  divisor,  27a' 

First  factor  of  correction,  9a +  4 

Correction  of  trial  divisor,  36a +16 

Complete  divisor,  27a'  +  36a  + 16 

according  to  formula  (a),  (JSJ37.) 


(172-175) 


CUBE   ROOT   OF   POLYNOMIALS. 


107 


CO 
1 

1 

1 

CO 

OS 

1 

CO 

OS 

+ 

+ 

"^ 

s 

-^ 

+ 

1 

+ 

2 

1-H 

+ 

1 

1 

5 

CO 


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r-( 

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m 

H 

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+ 

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w  T 

c«  + 


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s 

bo 


s  + 

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a 
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H 
CO 
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+ 

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r-l 

tH 

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j 

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(176) 


108 

I 

00 

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EVOLUTION. 


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•a 

e 

e 

<N 

cs 

1 

1 

*« 

« 

CO 

CO 

o 

CO 


cs    cs 

+  + 

cs    cs 


5 

+ 

1 

cs*^ 

1 

e 

10 

»o 

r-l 

+ 

+ 

SI 

c 

c* 

e 

e 

e 

cs 

1 

1 

CD 

1 

1 

•• 

1 

1 

8 

e 

"e 

CO 

CO 

CO 

(175) 


CUBE   ROOT   OF   POLYNOMIALS. 


109: 


-f  §■ 

§ 

> 

rO 

> 

5a 

i  "> 

^ 

s^ 

s 

^ 

+ 

^"^ 

4- 

+ 

+ 

1 

CO 

> 

:b. 

e 
1 

1 

1 

1 

1-i 
1 

1 

^ 

1 

1 

1 

1 

1 

1 

—  55 

Sa 

rO 

+ 

+ 

1 

•< 

4- 

1 

's 

5 

s 

> 

"« 

> 

1 

'8 

8 

1 

1 

+ 

V 

+ 

+ 

+ 

+ 

+ 

2a 

M 

3a 

'^ 

M 

'8 

CO 

M 

& 

1 

1 

1 

1 

1— ( 

1 

1 

1 

1 

1 

1 

1 

Sa* 

^ 

a 

I 

^ 

1 

's 

1-H 

CO 

CO 

CO 

CO 

+ 

+ 

+ 

+ 

+ 

1 

1 

1 

1 

§ 

1 

•a 

'8 

•^ 

'M 

^ 

1 

1 

^ 

I— 

1 

1 

1 

;§ 

c§ 

eg 

1 

1 

1 

1 

1 
3l 

1 

1 

+ 

+ 

^ 

;g 

+ 

+ 

+ 

r-4 

s 

s 

;§ 

c§ 

1 

1 

2l 
1 

+ 

CO 

+ 

CO 

+ 

^ 

CO 

CO 

•i»  "k 

M 

1 

^ 

<N 

O  c 

"« 

8 

CO 

S 

ft 
1 

1 

1 

1- 

+ 

+ 

+ 

1 

1 

1 

2  % 

"^ 

"^ 

1 

eg  ^ 

c<   1  •< 

'^i 

1 

rO 

rO 

rG     rO 

1 

1 

1 

*1»  o 

"h 

'M 

's 

I 

S 

+ 

+ 

^  ^ 

+  1 

+ 

+ 

+ 

+ 

+ 

rO 

f 

rO 

1 

% 

1 

1 

1 

1 

S 

l" 

1 

'^ 

% 

M 

5§ 

'M 

•^ 

S 

1 

1 

1 

+ 

+ 

f— ( 

1 

1— t 

1— t 

1—4 

I-" 

4. 

1 

1 

"8 

0 

^ 

CO 

+ 
1 

1-1 

1 

8 

1 

=1 

1 

8 

CO 

CO 

+ 

+ 

+    ' 

e< 

e< 

rO 

CO 

"8 

1 

1 

1 

w 

i 

1 

1 

35 

1 

1 

•8 
CO 

1 

1 

(176) 


no 


EVOLUTION. 


+ 
I 


«  % 


+      + 


+        + 


H 

H. 

H 

^ 

O 

-* 

CO 

CO 

1 

1 

1 

1 

1 

1 

1 

1 

"h 

^i 

H 

'h 

•«l 

w 

rn-* 

«»* 

o+« 

o*# 

H 

H 

+ 

+ 

*« 

« 

©< 

(N 

1 

1 

+ 
1 

•• 

"h 

"h 

i? 

CO 

00 

«»(* 

r-H 

-<*< 

-1 

+ 

+ 

+ 

-^ 

"h 

% 

<N 

T*« 

"* 

1 

1 

1 

« 

1 

1 

1 

H 

N 

H 

H 

CO 

CO 

CO 

CO 

CO 

■t 

+ 

+ 

^ 

"« 

1 

(M 

1 

i-t 

•V* 

1 

«(♦ 

«    N 


1        1 

1— t 
1 

S 

S  1 

1 

fe  ^ 

1 

+ 

+  + 

"•j^ 

^ 

'^  '^ 

r-t 

C^ 

-^ 

CO       CO 

O 

05 

CO 

Oi     Oi 

+ 

rH 

+ 

1 

+     4- 

i 

5 

1 

CO      CO 

f-4 

OS 

a 

a    S 

+ 

1 

+ 

1       1 

« 

cS 

CO 

S 

cs 

'^ 

:^    5 

J 

T 

+ 

1 

J 

CI     ^ 
1       1 

e 

»- 

u 

H 

>i 

H 

*« 

■^ 

HH 

c^ 

CI 

(N 

T 

CO 

1 

+ 

rH 

1 

1— t 
1 

S 

s 

S 

+ 

+ 

+ 

S 

«  « 

f-H 

+ 

+  + 

i       1 

« 

'n 

:i 

'^    '^ 

irH 

C<l 

f— 1    1— ( 

rt< 

+ 

1 

1       1 

+ 

-H 

14 

"«    '« 

eg 

C^ 

C<l 

d      CI 

r— ( 

f-H 

1— t       r-t 

+ 

+ 

+ 

+       + 

•« 

« 

« 

•« 

%       •« 

CO 

CO  . 

CO 

CO 

CO      CO 

^ 

's 

s 

+ 

r^ 

+ 

<a 

+ 

f^ 

S 

CI 

1 

1 

+ 
1 

(176) 


4^ 

1 

i 

1 

1 

+ 

1 

2l 

CUBE   ROOT   OF   NUMBERS. 


IM 


In  the  preceding  solutions,  we  have  made  a  formal  applicatio»  of 
the  rule  in  each  case.  This  will  enable  us  to  extract  the  cube  root 
of  any  algebraic  expression  by  direct  process ;  but  when  the  root  is 
a  binomial,  we  can  generally  find  it  by  simple  inspection.  Thus,  in 
the  first  example  the  cube  root  of  the  first  term  is  3a,  and  that  of 
the  last  term  is  4  ;  from  this  we  infer,  since  the  expression  has  but 
four  terms,  that  3a  +  4  is  the  cube  root.     This  is  easily  verified. 

Again,  in  example  4,  the  cube  root  of  the  first  term  is  a" ;  and 
dividing  the  second  term  by  3a*  we  have  +3ab  for  the  second  term 
of  the  root.  We  have  also  for  the  cube  root  of  the  last  term,  —6' ; 
and  dividing  the  preceding  term  by  36*,  we  have  +  3a6  for  the  term 
of  the  root  preceding  —6'.  Hence  we  should  infer  that  the  cube 
root  isa'  +  3a6— 6*,  which- might  be  verified  by  involution. 


CUBE  ROOT  OF  NUMBERS. 

(a43J5  page  1V9.) 
a*  +  3a'b  +  Sab'  +  b'  =  U8,8l1     (53,  Ans. 


3a  +  b               dab  +  b^ 

3a' 

153        459 

7500 

23877 

3a'  +  3a6  +  6'  = 

7959 

23877 

2. 


571,787  {8S,Ans, 
512 


119200         59787 
243  729119929         59787 

Note.— This  example  may  be  solved  by  inspection,  and  all  others 
in  "which  the  root  has  but  two  places  of  figures. 


3. 


183 


549 


1895       9475 


10800 
11349 


256,047,875   (635,  Ans, 
216 


40047 
34047 


1190700  6000875 

1200175  6000875 

(179) 


112 

EVOLUTION. 

4. 

354,894,912  (708,  Ans. 
343 

1470000 

11894912 

2108    16864|l486864 

11894912 

5. 

11,852.352  (22.8,  ^«*. 
8 

|1200 

3845 

62     124 

1324 

2648 

145200 

1204352 

668   5344 

150544 

1204352 

144,125,083,907  (5243,  Ans. 
125 


152 

304 

7500     19125 
7804     15608 

1564 

6256 

811200    3517083 
817456    3269824 

15723 

47169 

82372800   247259907 
82419969   247259907 

128,100,283,921 
125 

(5041,  Ans, 

1504 

6016 

750000    3100283 
756016    3024064 

15121 

15121 

76204800    76219921 
76219921    76219921 

8. 


127 


1412 


14166 


105,555,569,176  (4726,  Atu, 
64 


4800 
8895689 


41555 
39823 


662700 
2824665524 


1732569 
1331048 


|66835200 
84996166920196 
(179) 


401521176 
401521176 


CUBE   ROOT  I^F   NUMBERS. 


113 


9. 


731,189,187,729  (9009,  ^n5. 
729 


10. 


243000000 

2189187729 

27009    243081  243243081 

2189187729 

1,762.790,912  (12.08,  Ans, 
1 

300 

762 

32        64 

364 

728 

4320000 

34790912 

3608    28864 

4348864 

34790912 

11. 


1,061,520,150,601  (10201,  Ans. 
1 


302 

60^ 

30000 
[  30604 

61520 
61208 

30601 
12. 

3060] 

312120000 
L  312150601 

312150601 
312150601 

33,212,361.641,984  (321.44,  Ans, 

27 

92 

184 

2700 

2884 

6212 
5768 

961 

961 

307200 
308161 

444361             .  :: 
308161 

9634 

38536 

30912300 
30950836 

136200641 
123803344 

96424 

385696 

3098938800 
3099324496 

12397297984 
12397297984 

(179) 


114 


EVOLUTION. 


13. 


1,371,737,997,260,631  (111111,  Ans. 


1 

300 

371 

31 

31 

331 

331 

36300 

40737 

331 

331 

36631 

36631 

3696300 

4106997 

3331 

3331 

3699631 

3699631 

370296300 

407366260 

33331 

33331 

370329631 

370329631 

37036296300 

37036629631 

333331 

333331 

37036629631 

37036629631 

14. 


0.171,467  (0.55655+   Jin*. 
125 


15. 


155 

775 

7500 

46467 

A1  0*7  ti 

1655 

8275 

907500 
915775 

5092000 
4578875 

16655 

83275 

92407500 
92490775 

513125000 
462453875 

166655 

833275 

9257407500     50671125000 
9258240775     46291203875 

0.004,235,801,032  (0.1618,  Ans, 

1 

36 

216 

300 
516 

3235 
3096 

481 

481 

76800 
77281 
7776300 
7815004 

139801 
77281 

4838 

38704 

62520032 
62520032 

(179) 


CUBE   ROOT    OF   NUMBERS. 


115 


CONTRACTED  METHOD. 
(9^3,    page  181.) 


1^442249  +  ,  ^w». 
3.000000 

1 


300 

2.000 

34 

136 

436 

1744 

58800 

256000 

424 

1696 

60496 

241984 

62208 

14016 

432 

86 

62294 

12459 

6238 

1657 

4 

] 

8239 

1248 

624 

309 
250 

62 

69 
66 

1 1.912931+ ,  Ans. 
7.000,000( 
1 


300 

6000 

39 

351 

651 

5859 

108300 

141000 

671 

571 

108871 

108871 

109443 

32129. 

673 

115 

109558 

21812 

10967 

10317 

6 

5 

10972 

9875 

1098 

442 
329 

110 

113 

110 

(181) 


116 


EVOLUTION. 


16.38321261+,  Ant, 
156.000,000,000 
125 


7500 

31000 

153 

459|7959 

23877 

842700 

7123000 

1598 

12784 

855484 

6843872 

86833200 

279128000 

16143 

48429 

86881629 

260644887 

86930067 

18483113 

16149 

3230 

86933297 

17386659 

8693653 

1096454 

161 

16 

8693669 

869367 

869369 

227087 

173874 

86937 

53213 

52162 

8694 

1051 

869 

|32.643859  +,  Ant. 
34,786.000 
27 


92 

148 

2700 
2884 

7786 
5768 

966 

5796 

307200 
312996 

2018000 
1877976 

978 

391 

318828 
319219 

140024 
127688 

10 

3 

31961 
31964 

12336 
9589 

3197 

2747 
2557 

32 

190 
160 

3 

30 

28 

(181) 


CUBE   ROOT   OF   NUMBERS. 


117 


«. 


|2.222222    +,  Ans, 

10.973,937 
8 


1200 

2973 

62 

124 

1324 

2648 

145200 

325937 

662 

1324 

146524 

293048 

147852 

33889 

666 

132 

147984 

29597 

14812 

3292 

7 

1 

14813 

2962 

1481 

330 

296 

148 

84 

30 

111.44740066   -f,  Ans. 
1,500.101,520 


300 

600 

31 

31 

331 

331 

36300 

169101 

334 

1336 

37636 

150544 

3898800 

18557520 

8424 

13696 

3912496 

15649984 

3926208 

2907536 

8432 

2403 

3928611 

2750028 

393101 

157508 

84 

14 

393115 

157246 

39313 

262 

236 

39 

26 

23 

(181) 


118 

7. 


RADICAL    QUANTITIES. 


|1.051963  +,  Ans, 
1.164,132 

1 


305 

1525 

30000 
31525 

164132 
157625 

315 

32 

33075 
33107 

6507 
3311 

3314 

3196 
2983 

331 

213 
199 

33 

13 
10 

RADICAL  QUANTITIES. 
(^47,  page  183.) 

1.  V^=*^26x3=>/3,  Ans. 

2.  f^8^«=f'49a«  X  2  =  7af/2,  Ans, 

3.  V\2x''y=VAx'  x  3y=2a4^,  Ans, 

4.  V547*=V27j;'  x  1x=z^x\^2x,  Ans, 

5.  4^108  =  4^27  X  4  =  12V4,  Ans. 

6.  f^x'-aV=*^a:»(a;-a')-a:V^^3^,  Ans, 

7.  6^32^=  6 Vsa'  x  4:^12aV4,  Ans. 

8.  3f^2  8a V  =  3f/4a V  x  7a  =  QaaVia,  Ans, 

9.  Va»+a'6«=>/a'(l  +6')=aVr+6',  ^»«. 

(181-183) 


REDUCTION.  119. 


10.  {x-y)V2x*-^x'y  +  2xy''=i{x-y)V2x{x'-2xy^-y^) 

=  {x-y)V2xx{X'-yy=(x-yyV2x,  Ans, 


11.  (a  -  hy^d'h  +  ^ab^  +  26'  =  (a  -  6)f'26(a'  +  2ab  +  6') 

12.  66(6'-6')2=56[(6-l)6']^=56'(^-l)^  ^'i*- 

13.  (2a'6»-3a'6')5=[(2a«-36')a'5']^  =  a6(2a'-36y,  J»*. 

14.  ^(a*6'  +  a*6*)^=^[a»6»(a6'  +  a«6)]^=a'(a6'  +  a'6)^,  Am, 


15.  ♦^8a'"'a^=V'4a'V»  X  2  =  2a V'V2,  Ans. 

16.  V^^= Va^V'  X  ar^=a'^c"'y/a'^c'^'^^  Ans, 

1  11 

1 7.  (2a*»y-  -  3a*"y»'»)*'=  [a;»«y"(2  —  3a:"y«-)]-  =ar'y(2  —  3a-"y"»)"», 

1  1  1 

18.  a-*"c(a'^c'''— a'-^V)  "  =a-^c\ar''<r((r—a'^)]  '* =€*(€"— oT)  *, 

(348,  page  1.84.) 

s.VT^=\/jj^=vir;rio=^VTo,Ans. 

4.  >/Y=Vi|z=iV75,  Ans, 


5,  ^^=^H=^j*'t  X  6=5*T^6,  Ans. 

6.  2|/|^2|/|=|i/6^,  ^n,. 

(183-184) 


120  RADICAL   QUANTITIEa 


8.  -\  -T-;=-V  -T-i  x«*=-^^a^  ^W5. 

(049,  page  185.) 

3..  {a'^cz)=  [(a-c^)*]^  =  (a*-4aV2:  +  6aV«-4acV  +  cV)^, 

Ans, 

6.  (a— 26)V^=*^2a(a-26)'=*^2a(a'-4a6  +  46'') 


(950,  page  187.) 

4.  The  least  common  multiple  of  1,  2,  3,  and  4  is  12  ;  whence  by 
Rule  II, 

»Va",  *'V^,  ">/^  'V^«,  ^n5. 

8.  The  least  common  multiple  of  2,  m  and  n  is  27nn ;  whence, 
•"Va-^^V^,  '*V?y^,  ""V^^^V^,  Ans. 


ADDITION  OF  RADICALS. 


(051,  page  188.) 


1.          ^l6a*x=4aVx                            2.  ^32=  4*^2 

Vi^=2aVx  Vn=  QV2 

QaVx,  Am.  f^l28  =  _8l/2 

18*^2, -4««. 

8.          V40=   2V5                            4.  Vl08=   3V4 

Vl35=   3V/5  9V4 

^625=   5V5  Vl372=   7V4 

ioV/6,  ^n«.  19 V4,  Aim. 
(184-1S8) 


V. 


ADDITION. 

i; 

V^  =  ^V2 

6. 

Vv=  |V3 

V^=l^2 

^fi=   ^V3 

Vj\=iV2 

^fH=   |V3 

y2,  Ans. 

iiV3,  ^n^. 

Kl   =K6 

8. 

3Va6»i'=3mf^ 

if^=if/6 

mV4ab  =  2myab 

i^U  =  T\^^ 

V25abm'=5mVab 

lif/6,  Ans. 

10wVa6,  ^>w. 

121 


0.  2aVc*x--c'y=2acV'a:— y 

3c  Va'ar — a'y  =  3ac  l^a: — y 

5  ^a^c^x — d*c^i/  =:5acVx—y 


lOaci^a;— y,  Ans, 


10. 


K20a'm— 20acm  +  5mc'   =f^5m(4a'  — 4ac+c')      =(2a--c)>^6; 


V207»c'  — 60acw  +  45a'7»  =  V'5m(4c'  — 12ac4-9a')  =  (2c— 3a)i^6»i 


(c— a)l^5m,  ^rw. 


11.  3V^«  =  3a:Vc 

y/ax*=z  Xy/a 

2>^ax*z=2xVa 


12. 


3xv/c  +  3a;Va=3ar(Va4-Vc),  Ans. 
2  j:(a'c? — aV)  3  =  2ax(rf — c)  3  =  —  2aar(c — c?)^ 


i 
Zax(c—d)^^  Ans. 

Note. — In  transforming  the  second  quantity,  observe  that  the  cube  root  of 
a  negative  quantity  ia  negative. 

(188-189) 


122  RADICAL   QUANTITIES. 


'       ai-b         '    {a  +  by^  '     a  +  b 

Hence  the  sum  is, 

(4_+-L.+°±:^y.-rr6-=("'-°^+°^:+'''--'^*Va.JI' 

\a+b     a—b      a'—b' J  \  a'—b'  f 

=2^^*^^',  Am. 


"•       ♦'(^ +«)" -'^^S'('  +°)=TT^''i +« 


nee  the  sum  is 
Vl+a     l+a     1— a/  1— a'  1— a 


1-a 


1, 


SUBTBACTION  OF  RADICALS. 

(asa,  page 

189.) 

41^135  =  121^15 

2. 

♦^75=51^3 

^Veo  =  4V'l5 

♦/50  =  5*/2 

8*^15,  Ans. 

6(V^3-f'2),-4n*. 

(188) 

SUBTRACTION.  123 


3^^=      3aVb  i^^W=i^Hfll=,\*/lT 

(1 2a'  -  3ayb,  Ans.  |f/lT,  Ans, 


5.  ?i/l?_^-l'^V4^'.5^MV5 
S'^      338       3^^      169  39 

^V36l_W36r:_19a 
IS'^      5   ~13^    25  65 

Hence  the  difference  is 

(14a     19a\  ,        a   ,       , 
39       65/         16     ' 

6.  (aV-3c'a;)3=c(a'-3x)^ 
2(aV-3rf*x)3  =  2rf(a'-3a;)i 


(c-2rf)(a'— 3ar)3,  ^^j. 


1.  (a'-a6'  +  a*6-6y={o'-6'+a6(a-i)}^ 

=  {(a«  +  a6  +  6'  +  a6)(a-6)i^=(a  +  6)(a-fi)^ 

(a'-3a'6+3a6»-.6«)^=    (a-6)^  =(a-6)(a--6)^ 

1 
26{a— 6)2,  ^w«. 


8.       a/^=„*|/^=V^=A^-"^ 
a;  — 1  '^    ar  — 1  '^    (•^"'1)       ^""^ 

'     «  +  l  '   «  +  l         '^    (-e  +  l)      *+l 


~Vl^l,  An,. 

«'— 1 


(l89-i90) 


124  RADICAL   QUANTITIES. 

MULTIPLICATION   OF  RADICALS. 
(3535  page  191.) 

1.  6»/6  X  3*/8  =  15V^  =  15V^4  x  10=:30Vlo,  Ans, 

2.  4V'l2x  3^/2  =  12^24  =  12^4  x  6=24t/6,  ^n*. 

3.  3f^2  X  2f^8  =  6^^6  =  24,  ^n5. 


4.  2*^5  X  2*^10  x3f/6  =  l  2*^300  =  12M0ax3  =  120f^3,  Ans, 
6.  2^^14x  31/4  =  6^56  =  6^8  x  7  =  12V7,  Ans, 

6.  oc^/^X  cVa'  X  V^'  =  5cV^V  X  cVa*  X  V^=5c'  V^V 

7.  (xy)^  X  (^)*  +  (y^)^=(xy)^'^  X  (xV)tV  X  (yV)!"? 

8.  (:r-y)5 x (x  +  y)^=[(ar-yr(x+y)']^^='V(^-y)>+y)' 

= "^l(^-y)(^+y)}>+y) 

=  V(:r'-y')»(a:  +  y),  ^n*. 

9.  Vl5  X  ♦^10=^225  X  Vi000=V225000,  ^n«. 

a « /a;     yi/b'     j  /6?     ay  s  /x'      e  /5*     V^* 


DIVISION.  125 

11.  If  we  square  the  two  first  factors  under  the  radical  sign,  V'    , 
we  may  extend  the  sign,  ^/     ,  over  the  whole  product ;  thus. 


DIVISION  OF  RADICALS. 
(254,   page  192.) 

3     V20a^__  V400aV^  _  V  400aV'       V  16a 

4.  (^qi  =  (a«6')^  =  (a5)^  ^n.. 

^    (16a'-12aV)*      .^        ^  .\     . 

5.  ^ — '—  =  (4a— 3ar)'^,  Ant, 

(4a')^ 
^      45      3-5      3(5')^         .^    ■ 

^    12c'(a-;r)*      3c[(a-xyy^     ^  .         .  •       . 

8.   ^^ '—= — 5-^^ '-^^ — =:3c(a— ^)Ta,  Ant, 

4c(a-x)3  [(a_ar)']TV 

i-  JL 

9.  ^ ^  =  i — i-L_=  (^8»-"c— 3'«)«-,  Ans. 

{ac'y     (a'^c"")"^* 

(191-192) 


126  RADICAL  QUANTITIES. 


10. 


Vx     Va  Va' 


Va' 


Va'b-ab'     Va'b-ab'     Va'bWb-ab')      1    w-— ~ 

(a585  page  194.) 
1.    {V2^y=zV8^'=V8^,^^^=aV8^,Ans, 

3.    (3V4a^)*==:8lV266a»V==8ir  ^256a"c*==8lVi6a'V 

=  81  Vsa'  X  2oc'  =  1 62a'  ^^20?,  ^n*. 

5.  (Vl2^)'=t/l2^'  =  f^46'  X  3a=26V3^,  ^n*. 

6.  ("v/c(a_x)')'=(Vc(a-a;)')'=Vc»(a-ar)*  =  (a~a:)Vc'(a-ar), 

8.    (Vxy(x-y)y  =  Vxy{x-yy=xyVxi/{x-y)%Ans. 

10.  f-V96^')  =^  ^96^"=^  Vs2x'  xdcx=—V^^  Ans, 
\x  /      x'  x"  X  ^ 

(^595  page  196.) 


3.    y2v^98=y  v^8x98=:r  v/784=V28,  ^n*. 
(192-196) 


POWERS   AND   ROOTS.  127 

4.  Putting  the  fifth  power  of  the  coeflScient  under  the  radical  sign, 
we  have 

r   v^^^486:=y  v/64=V8,  Ans. 

6.  Y  5V 5 =yVU5=y>^  12~5=V 5^  Ans. 

11  J-  i 

~^j\       =  \-r-if      ;  dividing  out  the  factor  2  in  the  expo- 

(ax  \^ 
-^-\  ,  Ans. 

(aeO,  page  197.) 
8.  Vl296=V  Vl296=f/36  =  6,  Ans, 

4.  Vl7797851516625=y  ^17797851515625 

=V421875=76,  Am, 

6.  Vl9110297'6=y  ♦/I91102976==v^l3824=24, -4iM. 


6.  V65536=y  |/(65536)2=|/(256)^  =  (l6)i=4,  Ans. 

7.  We  first  take  the  square  root,  which  is 

a»_4a6  +  46'; 
then  the  square  root  of  this  is,  a— 25,  Ans, 

8.  The  square  root  of  the  expression  is 

and  the  cube  root  of  this  is  a*  +  5,  Ans, 

(196-197) 


128  RADICAL   QUANTITIES. 

GENERAL  THEORY   OF  EXPONENTS. 
(aei,  page  199.) 

6.  a*b*  xJb^=Jb^=a^y^,  Ans. 

7.  Adding  the  exponents  we  have 


aM       1-1      1-1       -»       1      /c'\i    ^ 


(jr*V     *^       ^ 


{x^y    x^ 


10.  (a^-a^)(a^  +  l)=a^4-a^--a^-a*=a*-a^,  u4n*. 


11.  {2Vx'+yxy){SVx-yxy)=eVx'-2Vx'-yzy  +  3Vx'yi^-xy 

=  62;— 2Va;y +  3Va;y— a:y,  ^n«. 

12.  a^-2a~*  +   a"*  ,     ,  ,        ,. 

13.  a--b      Va+Vb 


J-    a-i 


1 


a-hyablVa-^Vb,  Am, 


a*— 2a'      +    a  *  —Vab^b 

—  a"     +2a~*— o~^  — Va6-6 


a*— 3         4.3a  *  —a  ^  ,  ^W5. 


(199-200) 


THEORY  OF   EXPONENTS. 


129 


14. 


15. 


16. 


Ill 
J-  J 


a'-l 


a   —a",  Ans. 


-  J+J 


A  i.  1    A.  18  .!_• 

J-b^ 


4  2  i.         3  4         1    !_• 

a*  +  a^b^  -\-a'b^  +a*b'  +  ab^  -{-a^^ 

5  2.  i-  i  11    1J» 

-.a^b''~-a'b''-a*b'-ab^-a^b  =>  -6* 


a'—b*y  Ans. 


i.         Ill 


js+ar^a^+a^ 


2.       11       a. 


1       a.  a.       1 

If       i         F 
111        !• 


/Irirf^Sy     ld=31/5  4-15±6f/5     ^.    .        . 
\~2~/  "^ 8 =2±|/5,  ^w.9. 


19.  Raising  each  factor  of  the  numerator  to  the  power  of  nr^  and 
each  factor  of  the  denominator  to  the  power  of  ws,  the  expression 
becomes 

cT'y.cT'x  oT'  ~~  c'^+'-"«-  ~\c) 
(200) 


Ans. 


130  BADICAL  QUANTITIES. 

(  2f3-2Vl08  )  ^       j  2t/3-6V4  |  '     ft'  /i\^     /2  '  /72?\* 

•   llV^sWf   ^leWaFsl  =(31/9;  =l3V  ^3)^ 


*    (iV5  +  |V2)  (     i(2l/5  +  5V2)      f 


(21/5 +  5V2) 

=?  (  (2l/5-5V2)(2V5  4-5V2)  )  * 
~5(  2V/5  +  6V2  f 

=  |(2V5-5V2)^,  ^»». 


22.  Performing  the  multiplications  indicated  in  the  numerator 
and  denominator,  the  expression  becomes 

a-h 


Va-n 


—Va-\-Vhy  Ans. 


23.  Omitting  the  exponent,  the  numerator  is 

(f/5  +  2)(V5+i'2)(V5-»/2)  =  (*/5  +  2)(i/5'-2)  =  5-4  =  l; 
and  the  denominator, 

(♦'i3  +  3)(VT3+f^3)(VT^-3)=:(n3  +  3)(*1i-3)=13-9=4; 
and  taking  the  square  root,  i,  Arts, 


IMAGINARY   QUANTITIES. 

(  268,  page  203.) 
10. 

(a  +  V^c)*=a*  +  4ay=:^  +  6a'(»/~c)V4a(V'irc)'+(f/— )\ 

=a*  +  c'  — 6a'c  +  (4a*— 4ac)f^^,  Am. 

(200-203) 


QUADRATIC   SURDS.  131 

11.  a^V-^ci)     a'-f   ^^^a     {a-\-V —a—\^  Ans, 


a  — ay  — a 


aV^^a  +  a  ^ 

-a  +  ^/Hi 
12.  The  equation  can  be  written, 

From  (!807)  we  must  have 

a  +  y=c  +  ar,  (1) 

and  xVc=yVa,  (2) 

Whence,  by  substituting  in  (1), 

Vc 
X — —x=ia—c 

x=^ — —, —  =:a-\-yac  I 
In  the  same  way,  y=c-\-  ^ac  ) 

(275,  page  208.) 
4.  Here  wehave  a=ll,    Vb-QV2,     6  =  72;    hence, 


ll4-f^l21-72     ^ 
x=. =9, 


ll-*^21-72     „ 
v= =2 ; 

f/a;+|/y=3+i/2,  Ans, 
5.   o=7,    f6=4V'3,     6=48;    hence, 


7  +  *'49- 

-48 

~            2 

7-n9- 

-48 

y= =  3; 

y  2 

^x—yi/  =  2—yd,  Ans, 

(204-208) 


132  RADICAL    QUANTITIES. 

6.    a=7,    Vh=2VlO,     6=40;    hence, 


7  +  */49- 

-40 

2 

7-^9- 

-40 

2 

Vx--Vy=V5-V2,  Ans. 


Y.   a=94,    Vb=42V5,     6=8820;    hence, 

94  4- ♦^8836 -8820      _ 
ar= _ —40 


94 —  ♦^8836 -8820      , 
y= =45  ; 


Vx+Vy=1  +^45  =  7  +  SV5,  Ant. 


8.  a=28,    f^6=10V^3,     6=300;  hence, 

28+*^784  — 300 
x= =  55, 

28— ♦^784  — 300 

y= n =3; 


Vx+Vt/=5+ys,  Ans, 


9.  az^np-h  2m',    Vb = 2m*'njo  +  m',     6 = im'np  +  4m* ;  hence, 
^_n/?  -f  2m'  +»'ny  +  4m'n/?  +  4m*  —  4m^np— 4m* 

2 
=np  +  m''; 

_np  +  2  m'  — ^w'y  +  4m'n;7  +  4m*— 4m'np— 4m* 
y  _ 

=m' : 


f'a;— V'y=Vwp-fm' — m.  Ant, 


10.  a=6c,     f'6=26<^6c-6',     6=46'c-46*;  hence, 

6c+*^6V-46'c  +  46*     ^       „ 
ar= =6c-6', 

6c-i^6V-46'c  +  46*     „ 

y= 2 =^^ 


i^a?+f/y=*^6r— 6'  +  6,  ^n*. 
(208) 


QUADRATIC   SURDS.  133 


11.  a=Y,    Vb=30V-2,     6  =  - 1 800  ;  hence, 
7+*^49  +  1800 


2 

7—^49  +  1800 


=      25, 
=  -18; 


2 


12.  a=16,    f/6=30V^-l,     6=-900; 

16+^256  +  900 
:r=: =  25, 


16— K256  +  900 

y= 2 =~9; 

♦^ar+V'y=5  +  3VCri 
also,  i^ar-»^y=5-3»^iri 

10,         Ans, 

13.  We  add  the  answers  of  examples  4  and  6,  and  have 

3  +1^5,  Ans. 

14.  a=:31,    VA=12»^^,     6=-720;  hence, 


31+V961  +  720 
x= =   36, 

31-1^961  +  720 

y= , =-5. 

Again,  a=K—l,    V6=4f^^T,     &=— 80;  hence, 

—  l+»^l  +  80 
x= =     4, 


— i-vr+80 

y= 2 -=-'' 

Therefore,  6  +^^  +  2  +f^=8  +  2VC:6,  Ans. 

(208) 


134  BADICAL   QUANTITIES. 

RATIONALIZATION. 
(280,  page  211.) 

2.  Multiply  both  terms  of  the  fraction  by  |/6. 

3.  Here  the  factor  is  V«'. 

.    XTT    1  ^2x^81     ^2x\^2i~xS     V8xV9     ^72      , 

4.  We  have = = =— — -  ,  Ant, 

9  9  3  3^ 

5.  The  factor  is  f/7—f'3. 

6.  The  factor  is  j^a-\-\^c. 


,  Ans, 


7.  The  factor  is  VTT+f^5,  and  we  have 

11 4-2*^55  + 5  _8  +  »/55 
6  3 

8.  The  factor  is  VTT-V'3. 

9.  The  factor  is  VTo+*/6  ;  whence, 

4 

10.  Multiply  both  terms  of  the  fraction  by  t^6— V^^,  and  we 
have, 

6-2f^^irTg-3     1-t^^^^      , 

s =—4—^  ^'^*- 

11.  The  factor  for  rationalizing  the  denominator  is  (5  -\-\^5)(V3  —  1); 
whence 

(3+»^3)(3+»^5)(f5-2)(5+t^5)y3-l) 
40 
The  product  of  the  first  and  last  factors  of  the  numerator  is  2  f^3  ; 
of  the  others,  4V5.     Therefore, 

SVTE       ,/—    , 
=1V15,  Ans. 


40 

(211-212) 


RADICAL   EQUATIONS.  135 


12.  Multiply  both  numerator  and  denominator  by  1+a+^l— «'; 
whence, 

(l-i-«)'  +  2(l+a)»^T^^'  +  l  -a'     2(l+a)4-2(l+a)»^r^' 
(1 +«)'  —  !+«'  "■  2(l+a)a 


,  Ans. 


13.  Comparing  this  with  example  2,  (^TO),  we  have  a=5,  x=2  ; 
and  the  complete  multiplier  is 

(^/5  +  V2)(5  +V2)  =  5f/5  +VlO  +  5V2  +  V8,  An», 

14.  By  (3)  the  factor  is 

x*—y'      a*—b*        L       „i       14        ,,       18        u> 
--=-1 ^=a«+a'63+a263-fafe'4-a='63+6  3 

Hence,  j— ^^ ? ,  Ans, 


RADICAL  EQUATIONS. 
(a82,  page  214.) 

1,  Transposing  Vxy  and  squaring  each  member  of  the  equation, 

x-\-1=z4d—Uyx-\-x, 
UVx=iA2, 

Vx=Z  ;  ir=9,  Ans. 

2.  Squaring  each  member, 

a:'  +  6a;  +  9=ar'— 4a;  +  69, 

10^  =  50  ;  x=5y  Ans, 


3.  By  squaring,  Vx  +  48  =  2^x; 

squaring  again,  3ar=48 ;  x=lQy  Ans, 

(212-214) 


136  RADICAL  QUANTITIES. 


4.  By  cubing,  |/j;  +  2V^a  +  a;=a— ♦^a  +  a;. 


Squaring  this  result,  x  +  2Va  +  a; = a'  —  2aVa-\-x-{-  a+x, 
2(l+a)*^a+^=(l+a)a, 


a'  a'— 4a     ^ 

a-f-a;=— ;  «= — - — ,  AnS, 

4  4 

5.  Multiply  by  c  f'ar,  then 

ac  +  az=c  V7i;  or  x=c(Vn'-a\  Ans. 

6.  Multiply  by  f^l  —a:',  the  least  common  multiple  of  the  denomi- 
nators, and  we  have 


y^- -^ L  +  l^x=3x, 

1 — X 

(l—x)-\-(l+x)=3x;  hence  «= |»  -4n«. 


V.  Multiply  by  Vc-l-x, 


c+ar=Va4-p, 


squaring,  c'+2car+^'=a  +  ar;  hence  ^=""5 — , -4»«. 

2(7 


6.  Transposing  x  and  squaring, 


9  +  a^a;'— 3  =  9-6a;  +  a;', 


a;'  — 3  =  36  — 12ar  +  a;', 
12a:=39  ;  or  «=3{,  Ans. 


9.  By  transposition,  2V« — 32 = 2 (\^x—ys), 

ar— 32=a;  — 2V^+8, 
V^=20, 
8a:=400;  hence      a;=60,  ^n*. 
(214) 


Ans, 


RADICAL    EQUATIONS.  137 

10.  The  least  common  multiple  of  the  denominators  is  «— 4,  or 
(yx^1i){i/x-{-2) ;  multiplying  by  this, 

ayx-}-2a—a  —  c=c\^x—2Cy 
{a~cyx=—ai-c ; 

therefore,  ar  =  —  I  — —  J  , 

11.  Factoring  the  first  member,  we  have 

Vm(yx—l)_Vx  +  m 
?c(f^x-l)  —  Vx  +  c' 
VmX'\-eVrn=Vcx-\-mVc, 

therefore,  xz=mcy  Ans. 


12.  By  cubing,  a*'-da'x  +  x*Vda—x=a*'-3a*x-{-dax*—x'; 

V3a—x=:3a—Xf 
or  1  z=Vda—x ;  or  «=3a— 1,  Ans, 

13.  Clearing  of  fractions, 

xVc* — aa:  +  c' — ax = c*, 

c'—ax=:a' ;  or  «== ,  Ans, 

a  f^ 


14.  Squaring,    i+^-^l^+lf^, 


transposing,  <fec.,  -  + 

X     5  5     X 


1,41,4      1,1 
squaring,  _  +  _._  +  _=_4.- ; 


11^ 

-=^  ;  hence,  a;=20,  ^n*. 


15.  By  squaring,  a— a;=i^a+V 

a*—2ax-{-x*=a-{-x\ 


2aa;=a'— a  ;  hence,  ar=———,^«#. 


(214-215) 


138  RADICAL   QUANTITIES. 

16.  Clearing  of  fractions, 


VA-\-5x-[-x*z=zV4—Xy 


V4  +  5x-^z^  =  4—x, 
4-\-5x-{-x^  =  lQ  —  Sx  +  x*\  hence,  a;=||,  Ans, 

17.  By  transposition, 

v/5-x=ViO-V5+3-, 
cubing,  5-x=l0-  (3VIoo)(V5Ti)  4-  (sVlo)  (V(5+ar)')  -5-ar, 
canceling  terms,       (v^io)  C>/(5 T^)  =  (3 Vioo)  {VJ+x), 
dividing  by   (3VlO)(V/5  +  a?),      v/6+a:=Vio, 

5+x=10;  hence,  a: =5,  ^n«. 

Note. — If  we  had  transposed  the  term  Vs — x  and  solved,  we  should  have 
found  a;  =  —  5  ;  and  it  is  evident  that  either  value,  x  =  ±  5,  satisfies  the  origi- 
nal equation. 


18.  Multiply  hy^a+x, 


Vax  +  x*+a-{-x=2x, 


yax-\-x*=a—Xj 
tzx-\-x*=a*  —  2ax-\-x*^ 

therefore,  x=-jAn8, 

3 


19.  Squaring,  x^  +  2ax-\-a*=a^ -h^b*x'-\-x\ 

ir'  +  2aar=V6V  +  a:*, 
X*  +  4ax*  +  4a  V = 6  V + x\ 
4<M;  +  4a'=6'; 

therefore,  x= ,  Ans, 

4a 

20.  Clearing  of  fractions, 

24ar-8*^6x  +  6*^6^— 12=24a:-9*^+8*^6^— 18; 
whence, 

^6^=6;  x=6y  Ans, 

(216) 


RADICAL   EQUATIONS.  139 

21.  Cubing,  we  have 

4-f-z 
or,  64-\-x*—8z=l6  +  Sx+x\ 

I6xz=4:8;  hence,  x=df  Ans. 


22.  Multiplying  by  ^5  +^, 

VdX-^X^zzzlO  —  X, 

by  squaring,  5ar— ar'  =  100  — 20a;4-a?*, 

252:=100;  hence,  x=:4,  Ans, 

23.  Multiplying  by  the  denominator  Vx-\-Vx^ 

x+Vx-Vx^-x-^Vx', 
dividing  by  ^x  and  transposing, 


Vx-l=Vx-i, 
squaring,  x—l=x —Vx  +  j, 

-f/arcri;  hence,  x=\^^  Ans, 

24.  Clearing  of  fractions,  we  have 

3aa;  +  56f^— 36Vo^— 56'  =  3(Mr—25Va^  +  36Vai— 26'; 
whence  fai=36, 

aa;=:96' ; 

therefore,  a;=— ,  ^n5. 

a 

25.  Clearing  of  fractions, 
transposing  and  dividing  by  2, 


4t/4ar+l=5l^, 
Squaring,  6  4ar  + 1 6  =  1 00  j*, 

36ar=  16  ;  hence,  x=z^^  Ans, 

(215) 


140  QUADRATIC    EQUATIONS. 

26.  Clearing  of  fractions, 

95f/x=190, 

Vx—'l  ;  hence.  ar=4,  Ans. 

27.  Multiply  the  numerator  and  denominator  of  the  left  hand 
member  by  the  numerator, 

a  a?— a* 

Multiply  by  a,  and  take  the  square  root, 


yx+i^x-a=± 


Vx-a 


mnltiplying  by  Vx—a, 

squaring  and  transposing,  we  have,  T=^a!^ —,  Ant, 


PURE  QUADRATICS. 

(^865  page  21 Y.) 

,     5.  aar'  +  l  =  (a-a-)(a  +  x), 

By  expanding  =:a'— ar', 

or,  (a  +  l)a:'=a»-l, 

a;'=a— 1  ;  hence,  a:=±f^a— 1,  Ant'- 

6.  Clearing  of  fractions, 

3(a;  +  4)'  +  3(a:-4)»  =  10(a;'-16), 
or,  62:'  + 96  =  10a;'— 160, 

4a:' =  25  6, 
a;'=64  ;  hence,  a:=  ±8,  Ans, 
(215-216) 


PURE   QUADRATICS.  141 

7.  As  before,     Q{x  +  2y  +  6(x-2y  =  lS{x'-4), 
or,  12i;'  +  48:=13a:'  — 52, 

a;'=:100;  hence,  ir=dblO, -4«5. 

8.  Clearing  of  fractions, 

(a;  +  a)'-f(ar-a)'  =  7(a;'-a') 
or,  2a;'  +  2a'=:7a;'  — 7a», 

5x*  =  9a\ 

,     9a'  3a       . 

X  = — ;  hence,  ar=  ±-r-- ,  ^ns, 
5  r  5 

2!  q 

9.  Omitting  -  and  -  on  each  side  of  the  equation  it  becomes 

l_x 

^""12'  _ 

or,  «'=12;  hence  x ±V I2y  Ans. 

10.  This  equation  has  the  same  form  as  example  9;  but  as  we  can 
not  unite  the  letters  as  we  did  the  numbers  in  9,  we  clear  of  frac- 
tions; whence, 

ex*  -f  CL*c = ax*  -\-  ac*y 
x*(a—c)=ac(a  —  c), 

x^=iac\  hence  ir  =  ±V^,  ^n*. 

11.  Clearing  of  fractions, 

«'  — 8  =  6  +  6f^5, 
whence,  ar'=14  +  6f^5 ; 

andby(ar5),  .  x=z±(d+V5),  Ans, 

12.  Multiply  by  6,  then 

2x*— 4f/2-3a;»  +  9  =  6-6f^2; 
whence,  a;'=3  +  2l/2; 

and  by  (  375 ),  x=db{l  +f^2),  Ans. 

13.  This  equation  may  be  written, 

whence,  ar"=:9;  or,  «=±3,  Jn«. 

(217-218) 


142  QUADRATIC   EQUATIONS. 

It  is  evident  the  the  value  a:r=--2  will  satisfy  the  equation.     If 
we  clear  the  equation  of  fractions  it  will  be 

a:'  +  2x'~9J--18=rO, 

a  complete  cubic  equation  in  which  x  has  the  three  values, 

ar=  +  3,     ar=— 3,    a:=  — 2. 

14.  Clearing  of  fractions  and  uniting  terms, 

6z'=6, 
ar'  =  1.2;  hence  a;=dbl.095445  +  , -4n5. 


15.  Transposing,  we  have 
(x-b)(x  +  b) 


=  -H(^-4), 


ar  +  4 
or,  12(j;»-25)  =  -ll(a:»-16), 

23i:*=476;  hence  «=  ±4.64924  +  ,  An8. 


AFFECTED  QUADRATICS. 
(aOO,  page  220.) 

The  answers  to  the  fourteen   first  examples  can  be  written  down 
immediately,  according  to  the  rule  under  (^890). 

15.  Divide  by  3,  thus 

a;'— 5x=:  — 4  ;  whence  ar=4  or  4-1,  An8» 

16  and  17  are  reduced  to  the  general  form  by  division  and  trans- 
position. 

18.  Multiplying,  we  have 

6«"— 6;r-.10a;+10=:2a:*  +  30, 
4ar'-16ar=20, 
or,  a:'  — 4ar=5;  x^zzb  or  -^1,  Ans, 

(218-220) 


AFFECTED   QUADRATICS.  143 

19.  This  reduces  to 

lOx'  — 16a;  +  10a:— 16=:5a;'  +  4ar  +  5a;  +  4, 
6a;'— 15ar=20, 
or,  x*—dxz=z4:  ; 

whence,  a: =4,  or  —1,  Ans, 


20.  We  have      9a:'  +  24ar4-16  =  54a: 
or,  X  —  3  X —       J  , 

whence,  ar=  y  ±  1  ;  ar=|  or  |,  Ans, 


21.  This  is  already  of  the  general  form,  and  we  have 

ar=  +Tj=^U  ;  whence  ar=|  or  —j\,  Ant, 

22.  Dividing  by  15, 

,     2a-     1 

a:=— j*j±f|  ;  whence  ar=|  or  —f,  Ans, 

23.  Dividing  by  4, 

,     13a?_  5 
*       28  "72* 
ar=i^±W'V  ;  whence  x=j\  or  — /j,  ^n^. 

24.  Multiplying  by  4(j:''  — 1),  the  least  common  multiple  of  the 
denominators,  we  have 

2(a:+l)  +  12=a;»  — 1, 
or,  a;'— 2a:=15  ;  whence  x=5  or  —3,  Ans, 

25.  Clearing  of  fractions, 

4(aj  +  2)(ar  +  3)4-5(a:+l)(ar  +  3)  =  12(a:  +  l)(ar  +  2); 
or, 
4ar'  +  12a:  +  8ar  +  24  +  6«'  +  16a?  +  5a;4-15  =  12a;'  +  24ar-M2a?  +  24. 

3ar'— 4ar=:15, 
ar'  — Aa;=   6; 
whence,  ar=3  or  — |,  Ans, 

(2»1) 


144  QUADRATIC    EQUATIONS. 

(a91,  page  223.) 

1.  Multiplying  by  5,  and  adding  2',  the  complete  equation  is 

25j:'  +  20.r  +  4=1024, 

52:  +  2r=±32;  x=Q  or  -^j*y  Ans. 

2.  Multiplying  by  5,  and  adding  2', 

25x'  +  20^  +  4  =  1369, 

5x  +  2  =  rt37;  x=1  or  ~y,  Ans, 

8,  Multiplying  by  V,  and  adding  10', 
49jr'~1402r+ 100  =  324, 

7a?— 10=  ±18;  x=1  or —^^  Ans, 

4.  Multiplying  by  24,  and  adding  16', 

144x'  + 3602: +  225  =441, 

12jr  +  15  =  ±21;  x=^  or  ^3^  Ans, 

5.  Multiplying  by  8,  and  adding  5'. 

16x'  — 40j:  +  25  =  961, 

4i:--5=±31;  ar=9  or  -  y,  ^n*. 

6.  Multiplying  by  21,  and  adding  146', 

441a;'-292x21a:  +  21316  =  10816, 

2l2r-146=±104;       ar=llif  or  2,  ^n.f. 

*!,  Multiplying  by  24,  and  adding  13', 

144a;'-13x  24j:+  169  =  25, 

12.r  — 13=±5;  ar=f  or  f,  ^n*. 

8.  Multiplying  by  28,  and  adding  3'. 

1 96a:' ~842r  + 9  =  4489, 

14a:— 3=  ±67;  xz=  5  or  —^^,  Ans. 

9.  Multiplying  by  12,  and  adding  53', 

36a:'  — 53  X  36j-  + 2809  =  2401, 

6j:-53=±49;      a:=l7  or  f,  ^n». 
^223) 


AFFECTED   QUADRATICS.  145 

10.  Multiply  by  4,  and  add  13', 

4a:'  +  52j:  +  169  =  729, 

2jr  +  13=:db27;  x=1  OT --20^  Ans, 

11.  Multiply  by  3,  and  add  4', 

9j:'  —  24:r  + 16  =  31 +  124/3, 
3^-4  =  ^(2  +  3^/3), 
whence,  ar=2+f3  or  |—f/3,  u4n«. 

12.  Multiply  by  4,  and  add  IT, 

4j:«  +  44jr  +  121=441, 

2x  +  ll  =  =t21  ;  x=5  or —le,  Ans, 

13.  Clearing  of  fractions  and  uniting  terms, 

2l2:»-292x=  — 500, 
which  is  example  6. 


14.  Clearing  of  fractions  and  uniting  terms, 

a:'— \7ir=-252, 
4a:«-148jr+ 1369  =  361, 

2a;— 37=  ±19;  «=28  or  9, -4n«. 

15.  Clearing  of  fractions  and  uniting  terms, 

2.r'--13:ir=— 6, 
16a:'  — 104a;+ 169  =  121, 

4a:— 13  =  ±11;  a:=6  or  ^, -4n«. 

16.  Clearing  of  fractions  and  uniting  terms, 

4a:'-lla:=-7, 
64a;*-l76a:  +  121=     9, 

8a;— 11  =  ±3;  xz=i  or  1^  Ans, 

17.  Clearing  of  fractions  and  uniting  termp, 

21a:'  — 94a:=-13, 
441a:'-94x21a:  +  2209  =  1936, 

21a:— 47  =  ±44;  a:=J/ or  |,  ^/la. 

(223-224) 


146  QUADRATIC   EQUATIONS. 

(aOS,    page  225.) 

1.  Here  we  have  ^=|,  and 

4a:  +  f=±l;  a?=i  or  —\yAns, 

2.  /zzrV'^; 

whence,  ar = 1 1  or  —  ^V»  -4««' 

3.  <=|; 

whence,  a:=|  or  ■^\y  Ans. 


4.     <=^; 


49  ,     6ar      9      , 


whence,  ^=if  or  —  il*  -^^*» 

5      t^5 ' 

IH^'-¥^  +  25=^36, 

whence,  «= W  or  —Hy  ^»«- 

6.  Clearing  of  fractions  and  uniting  terras, 
49x'  — SOjr^-Se. 
Hence  <=S% 

whence,  x=2  or  —  i|,  -4««. 

(394,  page  226.) 

1.  Put  2a=7,  and  we  have 

x^—20x=2a  +  l, 
ar*— 2aa:  +  a' =a' +  2a  + 1, 
a;— a=±(a  +  l), 

a;=2a  +  l,  or  —1 ; 
whence,  a:=:8  or  --l,  An9, 


AFFECTED   QUADRATICS.  147 

2.  Put  2a=ll,  and  add  a' ; 

x*  +  2ax  +  a' = a'  4-  4a  +  4, 
a:  +  a=±(a  +  2), 

a;=-(2a  +  2)  or  +  2  ; 
whence,  a;=~13  or  2,  ^»«. 

3.  Put  2a =17,  and  add  a* ; 

a;'  — 2a  +  a'=a'  +  6a  +  9, 
a:-a=±(a  +  3), 
a:=2a  +  3  or  —3; 
whence,  ar=20  or  -3,  Ans, 

4.  Put  2a=21,  and  add  a* ; 

ar' +  2a  4- a' =«' +  4a  +  4, 
a;  +  a==b(a  +  2)  ; 
whence,  ar=2  or -23,  ^n*. 

5.  Put  2a =75,  and  add  a' ; 

a;'  — 2ac  +  a'=a'4-2a4-l, 
ar-a=±(a  +  l); 
whence,  a^='^6  or  -1»  ^'W- 

6.  Put  2a =72,  and  add  a' ; 

a:' +  2aa;  +  a' =a' +  10a  4- 25, 
a;+a=±(a  +  6); 
whence,  a:=5  or  -77,  ^W5. 

7.  Put  2a =325,  and  add  a' ; 

a;'— 2a«+a'=a'  +  20a  +  100, 
x-a=±{a +  10); 
whence,  a;=335  or  -10,  Ans. 


(228-227) 


148  QUADRATIC   EQUATIONS. 

EQUATIONS  IN  THE  QUADRATIC  FORM. 
(«965  page  232.) 


In   solving  these 

examples,  we  shall  put  y 

equal  to  the  lowest 

power  of  the  unknown  quantity. 

1.  Put 

y=^'; 

y«_34y=-225, 
y«— 34y  +  289=      64, 

y=    n±8. 

x*  =  i/=.     25,  or  8 

•; 

whence. 

a;= 

=  ±5  or  ±3,  Ans, 

2. 

y=*'; 

y«-35y=-216, 

4y' 

'-140y4-1225=:      361, 
2y-35=±19, 

x*=y=     27  or  8 

» 

whence. 

ar=3  or  2,  Ans, 

3. 

y=^'; 

y'-4y  +  4  =  625, 
y-2  =  =b25. 

a;'=y=:27  or  - 

■23; 

whence, 

x= 

:3  or  V-.23,  Ana, 

4. 

y«'; 

y'  +  31y  =  32 
4y'-f31y  +  961  =  1089, 
2y  +  31  =  ±33, 
^•=y=l  or-32  ; 

whence. 

x=l  oPiC,  Ans. 

5. 

y-^; 

y'-y=56, 

4y'-4y  + 1  =  225, 

'■ 

2y-l  =  ±15, 
a;^=yz=8  or  —17, 
a:^  =  2orV_7; 

whence, 

(232-233) 

r=4  or  V49,  Ans. 

AFFECTED   QUADRATICS.  149 

6.  y=af; 

y'-2y+l=9, 

y-i  =  ±3, 

af=y=4  or  —2; 
whence,  a;=V4  or  VTr2,  ^n«. 

J. 

20/-31y=  — 12, 
1600y«-2480y  +  961=     1, 
40y— 31=±1, 

ar'^=y=A  or -1; 
whence,  a;=  (f)-,  or  (f )••,  ^n«. 

8.  y=Vx; 

3y«-103/=-3, 
36y'— 120y  +  100=     64, 

Va:=y=     3  or  | ; 
whence,  a:=27  or  ,^,  -4n«, 


9. 

y=»'^+ 

5; 

y*-y=6, 

4y'-4y4-l  =  2525, 

whence. 

Va:  +  5=y=3  or  —2; 

ar=:4  or  - 

-1, 

Ans, 

10.  y= 

=  (^+12)^ 

1 

y'+y=  6, 

4y"  +  4y  +  l  =  25, 

whence, 

(a;  +  12)^=y=   2  or  -3, 
a;+12  =  16or  81; 

ar=4  or 

69, 

Am, 

11.  y= 

=(a.  +  a)^; 

(x+ay=y=b  or  —36; 
whence,  ar=6*— a,  or  816*— a,  Ans, 

(233) 


150  QUADRATIC   EQUATIONS. 

12.  Transposing  x  and  squaring, 

or  «'— 21ar=— 64, 

which  gives  ar=18  or  3,  Ans, 

13.  y=V9x  +  4; 

y'  +  2y  +  l  =  16, 
V9x  +  4=y  =  S  or  —5, 

9j;  +  4=9or25;         x=^  or  I,  Ans. 

14.  y=Vio+«; 

y'-y=  2, 

4y'-4y-l=   9, 
Vio  +  a;=y=   2  or  —1, 
10  +  a;       =16  or      1  ; 
whence,  x=6  or  —9,  Ans, 

16.    y={x-5)^; 

y•-3y  +  f=J-|^ 

3. 

(i— 5)*=y=8  or  —5, 
ar— 5  =4  or  v^(-6)' ; 

whence,  ar=9,  or  5  +  ^{—o)\  Ans, 

16.  y=(l+ar-ar')^; 

16y'-8y4-l=     i, 
whence,  y=     i  or  — ^. 

Hence,  l-f  a;— a;'=|,  or  l+x—x*=:-^^j, 

x'-x  +  }=Uy  x'-x-^i  =  \\ 

x-r=:±,±Vn,  ar-i  =  db-it/lT, 

whence,  x=±±iVli,  x=idbi^Tl,  Ans, 

(233) 


AFFECTED   QUADRATICS.  *  151 

17.  y=Va;  +  16; 

y=f±i, 

'^a?+16=5  or  —2; 
Whence,  x=9  or  —12,  Ans. 

1j3.  Clearing  of  fractions  and  uniting  terms, 

81a;*  — 82a;'=  — 1, 
Six*— 82x'  +  -!-fi-»-=     J-f^Ji-, 

whence,  a;=dbl  or  ±i,  ^ws. 

If  we  add  1  to  each  side  of  this  equation,  we  shall  have  perfect 
squares ;  thus, 

81a:' +  18 +3=   100, 
or 

and  9ar  +  -=dbl0; 

X 

hence  two  quadratics  for  ar,  and  the  same  values  as  before. 


1 9.  Clearing  of  fractions  and  uniting  terms, 

225a;*  — 901a;»=:-4, 
226ar*-901a:'  +  iU_._8_oj.=AYJ._«j.,. 
15x*  =  9J>^^db3-Y^ 
x'=4:  or  ^ij  ; 
whence,  x—dt.2  or  iy'j*  -^^*« 

Here,  if  we  add  |  to  each  side,  we  have 
^^  ,     20  ,    4       961 

2        .31 

whence  two  quadratics  for  x, 

(233) 


152  QUADRATIC   EQUATIONS. 

20.        x*-\-2x'-1z^  —  8x-{-l2     {x*+x 


2a;*  4-   x 


—  8a;'— 8a?, 

Hence,  by  taking  the  square  root  thus  far  we  see  that  the  ex- 
pression may  be  written, 

(a;'  +  a;)'-8(a:'4-a-)  =  -12; 
or  if  y=a:*  +  a',  y'  — 8y=:  — 12, 

y'-8y  +  16=     4, 
y-4=±2. 

Hence,         a:'— ar— 4=     2,  or  a;'  +  a;— 4  =— 2, 

x'-\-x+{=    V-,  a;'  +  a:  +  i=     |, 
ar  +  |=±f,  a;+i  =  ±f, 

and  a;=2  or  —3,  ar=l  or  —2,Ans, 

21.  Multiplying  by  a*,  the  equation  is 

a;*  — 8ar*  +  19a;'— 12a;=:0,     (a;'— 4a? 
.a;* 


2a;'— 4a;  —8a;' 

—  8x'4-16a;' 


Hence, 


Sa;*— 12x, 


(a;'-4x)'  +  3(a;'-4x)        =0, 
(a:«_4a;)'  +  3(x'^4a;)  +  |  =  |, 

a;'  — 4a;=0  or  —3; 
ar»-.4a;  +  4  =  4,  a;'-4a;-f  4  =  1, 

and  a;=0  or  4,  a;=l  or  3, -4»#. 


22.  a;«-10a;'  +  35a;'-50a;+24=0,     (a;'— 5a; 

a;* 


2a;'— 5a;         —  10a;' +  35a;' 
—  10a;=  +  25a;' 


10a;'— 50a; 
(233) 


AFFECTED   QUADRATICS.  153 

Hence, 

(a;'-5a:)«  +  10(x«-5ar)  =-24, 

{x^—5xy  -\-10(x*  ^5x)  +  25=     1, 

x^  —  5x=  —  4:  or  —6 ; 

and  x=l  or  4,  a;=2  or  3,  An». 

23.  ar*  — 8aa:'4-    8aV  +  32a'ar— 9a*  =  0,  (a;'— 4aa; 

a;* 


2x'  — 4aj;  —  8aa:»+   8a  V 

—  8aa;'  +  16aV 


—    8aV  +  32a'a:. 

Hence,  {x* — 4cp;)' — 8a' (a;'  —  4ax)  =  9a* 

(a;'  —  4ax)'  —  8a'(ar'  —  4a;r)  +  1 6a*  =  25a* 

«'  — 4aar  =9a"  or  —a* 

a;'— 4aar  +  4a"  =  13a'     or  a?'  — 4aar  +  4a'=6a' ; 

and  ar=a(2±^13)     or  a;=ia(2dbV3),  ^»». 

24.  Performing  the  multiplication  indicated, 

y*-2cy'  +  cy-2y«  +  2cy=c«, 
or,  (y«-cy)'  -  2(y'-cy) =c«. 


whence  y =i  ± /|- 4- 1  ±*^r-f?\  ,  An«, 

PROMISCUOUS  EXAMPLES  IN  QUADRATICS. 

1.  Completing  the  square,  we  have 

4a;' +  44a; +  121  =441, 
2a;  +  ll  =  i:21; 
whence,  a;=5  or  —16,  Ans, 

(233-234) 


154  QUADBATIC   EQUATIONS. 

2.  Clearing  of  fractions  and  uniting  terms, 

whence,  x=4  or  — 1,  Arts, 

3.  Clearing  of  fractions  and  uniting  terms, 

whence,  x=2  or  —3,  Ans. 

4.  Multiplying  by  8(ar'  — 1),  the  least  common  multiple  of  the  de- 
nominators, we  have 

12  — 2a;  +  2=a:»— 1, 
or,  ar'  +  2ir  +  l  =  16; 

whence,  xz=S  or  —5,  Ans, 

5.  Clearing  of  fractions, 

{2x*+x—5)(x  +  l9)  =  (x^-\-4x  +  3){2x  +  l5), 
or,  16x*—52x=:li0; 

ICar*— 52ar  +  iA»^=ip, 

whence,  ar=6  or  —  f,  Ans. 

6.  Clearing  of  fractions, 

or,  ai'  +  27a;=     28, 

x*+2lx-\-i\a.=     ifJL^ 

whence,  a;=l  or  —28,  Ans, 

Ex.  Y  and  8  need  no  solution  here. 

9.  Add  mn  to  both  sides,  (  393  ), 

mx* — 2  mxVn  +  mn  =     nx', 

Vm  •  x—^mn = dc^n  '  x ; 

,  Vnm        . 

whence,  x=.     ,    .  ,  ^n*. 


AFFECTED   QUADRATICS.  155 


10.  Add  A  to  each  side,  ( 293 ) ; 


4^     8^     4_     64 
49      21     9~     ~9' 


whence, 

:=1 

or  -111, 

Ans, 

11.     Add  36  to  each  side,  (  293 ), 

x'       \2x       ^ 

^-6  =  ±2, 

whence, 

x=' 

162  or  76, 

Ans, 

12.  Transposing, 

,  we  have 

16      -       «         ,1- 

0, 

^ 

(2a;-4)*     (2a;-4)''^ 

^            1_ 

0, 

(2^-4)'        - 

(2.:-4)'        = 
2x-4c           =: 

4, 
±:2; 

whence, 

x=zd  or  1, 

-4n5. 

13.  Put  y^Vx^ 

+  11; 

y'+y=42, 
y'+y+i=-^P» 

y=6  or  —7, 

ar'  +  ll  =  36  or  49; 

whence. 

x= 

:±5, 

,  or±f^, 

^»5. 

14.  Add  5  to  each  side  of  the  equation,  and  put  y=:Vd;'— 2a;  +  5  ; 
then, 

y'  +  6y=16, 
y*  +  6y4-9  =  25, 

y=   2or-8; 
ar"  — 2a:  +  5  =  4,  or  ar'  — 2ar  +  5  =  64, 

ar«-2a:+l=0,  a:' —  2a:  +  1  =  60, 

whence,  ar=l,  a?=:l±24^15,  Ans. 

(234) 


156  QUADRATIC   EQUATIONS. 

15.  Add  JyV-c'  to  each  side,  (^93). 

x'  +  -'^'-x*+-\Y-^' = \Y^'  4-  34ar  + 1 6, 


xxcuuc, 

a:'=     4,  or 

a:'  +  V^=-4, 

and 

^=±2, 

ar=  — 8,  or  —  1,  ^n*. 

16.  This  equation  may  be  "written, 


._        2(^^x  +  l). 


dividing  by  Vx-J-l,  we  have 


whence,  x=zl  or  4,  .4n». 

17.  Clearing  of  fractions, 

2x  +  2Vx=      16— a-, 
^x-{-2Vx=     16, 
36ar  +  24V'a:  +  4=    196, 
6f';r  +  2  =  dtl4; 
whence,  ir  =  4  or  7|,  Jn«. 

18.  By  squaring, 

\  2f/2/  f/g     ' 

or, 

ar'(12-a;')_(2  4-ar')* 
2f^2  2V2     ' 

Hence,  x*—4tx*=  —  2y 

z*=     2^=1^2 ; 

and  «=±(2dbf/2)^,  ^n«. 

(234-235) 


AFFECTED  QUADRATICS.  157 

19.  This  equation  may  be  written, 


^     ^  '  X  ^  X  X 

which  is  divisible  by  y  ;  whence, 

/ — r    y^^-  4  ^^ 

Bysquaring,    a;  + 1-2^:^+^1  +  1==^=;      ^^1: V;^^/-/^^t/X^   -^ 

squaring  again,  and  clearing  of  fractions,      /       "yi  ^-   X   —  '' 

4a;' +  4x' =a;*  +  2j;' 4- 3j;' +  2a;  + 1, 
or,  a:*  — 2x«— a:"  +  2a;  +  l=0;  /^y'^"  -^  f^J^ 

taking  square  root,         a;'— a;— 1  =0,  ^/  -  /  /  (/j^ 

ar-i^rfciV'S;  ^ -- t  l/TtW^ 

whence,  x=\{\±Vb\  Ant, 


20.  Multiply  both  terms  of  the  first  fraction  by  1  +4^1— a;',  of  the 

second  by  1— f^l— ar' ;  then 

l+t'l-ar'     l-f'l-a:'     VZ 

ar'                    a:'              a:" 

or,                                                     2f'l-ar«=V3, 

squaring,                                                       x*=\\ 

whence. 

a;=d=|,  Ans, 

21.  This  may  be  written, 

(U/)'-T' 

/    1    \^     ♦^2a: 
U+a:/    ""  12 

1          2a? 

l+ar^Hi* 

«»  +  a?=72, 

4a:' +  4ar  + 1=289, 

whence,                                           2a;  +  l  =  d:l7; 

a;=8  or  —9,  Am. 

(285) 

158  QUADRATIC    EQUATIONS. 

22.  Multiply  both  terms  of  the  fraction  by  x+Vx^—Qy  and  take 
the  indicated  square  root ;  then 

x-h^x'-9 


f^a:'— 9  =  2a;— 6, 
a:*-9  =  4ar'-24a:  +  36; 
whence,  a:'  — 8a:  =  —  15, 

ar»_8a;  +  16  =  l; 
and  ^=5  or  3,  Ans, 

23.  Completing  the  square,  we  have 

x^        =27  or  -28, 

x^       =3  or  v/- 28;    

whence,  a; =243  or  V(_28)',  ^rw. 

Since  the  exponents  are  odd,  we  may  write  the  last  value  of  a? 

A  L  1 

thus,  —(28)*  =  -(2  X  2  X  7)=' =  -(32  x  32  x  iy 


=  -(512  X  2  X  Vy  =  -8(2  X  7') 


^^^- 


1 

t5\3 


1 


=  -8(33614)^ 

24.  Completing  the  square 

144a;"»— 312a:"  +  169  =  25, 
12a:"— 13=  ±5; 
whence,  a:=V|  or  V|,  ^iw. 

25.  Transposing  2a:  aT)d  squaring,  we  have 

2  +  2a; = c'  —  2c'a: — 4ca:  +  c^x*  +  4ca;'  +  4a:', 
or,  (c4-2)V-2(c  +  l)'a:=2-c«; 

by  (a»3),       (^  +  2)V-2(c  +  ira:  + |^;=(g^', 

^         ^        c  +  2  c  +  2  * 

c'— 2 
whence,  a:=l  or  .        .^,  Ans, 

[C  +  Z) 

(235) 


AFFECTED   QUADRATICS.  159 

26.  By  expanding, 

{a  +  xy=     x*-\-5x*a+l0x'a''  +  l0xW  -\-5xa*  +  a\ 
-  (a— a:)»=  —x'  +  5x*a  —  10x^a^  +  10xW  —  5xa*-\-a\ 

Therefore  we  have 

10aa;*  +  20aV  +  2a*=352a', 

x*+    2a  V=   35a*, 

x*-^   2a V+  a*=   S6a\ 

x^i-   a'r=±6a'; 

whence,  x=  ±:a V5y  or  ±aV—1^Ans. 

27.  Clearing  of  fractions  and  uniting  terms,  we  have 

a*cx^ — a(6 + 2c)x  =  —  (6  +  2c)  ; 
(A  +  2c)«_6'-4c« 


by  (a03),        a'cx^-^aib  +  2c)x+- 


4c  4:C 


whence,  x==—(b+ 2c±»/6'— 4cO,  ^n«. 

28.  Clearing  of  fractions, 

abx =(a  +  b-\-x)bx-^(a  +  b-{-  x)ax  +  (a  +  6 + x)ab ; 

multiplying  and  arranging  terms, 

{a  +  b)x*-{-(a-^byx=-^ab{a  +  b), 
x*  +  (a  +  b)x=  —  ab, 

a+b     a—b 

«=-- 2-±-2-; 

whence,  xz=z— a  or —b^  Ans. 

29.  Clearing  of  fractions  and  transposing,  we  have 

X*  +  8ar»  + 1 6x'  —  a  V  —  Sa'x — 1 6a'  :=  0. 

Extracting  the  square  root  according  to  (!896),  this  equation 
roay  be  written, 

{x'-{-4xy-a'(x  +  4y=0, 
or,  (ar'  +  4ar)'=a'(;r  +  4)', 

x^-\-4:X=±:a{x-\-4). 
Dividing  by  a;  +  4  a;=dba,  Ans, 

(235) 


160  QUADRATIC   EQUATIONS. 

30.  Multiply  both  terms  of  the  left  hand  member  by  the  cUnoiRU 

2a  ~  2a 

Omitting  the  factor  2a,  and  squaring  the  left  hand  member, 


2x— 2Vx'  — a''=a?, 

3    ' 
whence,  x=zdo2a^^\^  Ans, 

31.  Dividing  out  the  factor  a—x  in  the  second  term,  and  clear- 
ing of  fractions,  we  have 

a' -f  x' +  a' +  2aa;  +  a;' = 4a' +  4az, 
OP,  x'— aar=a', 

ar«-ax+-=-a'; 
whence,  a;=-(l±|/5),  Ans, 

it 


32.  Clearing  of  fractions  and  transposing,  we  have 


)/2ax  +  x^  =ah  -|-  bx— a—x\ 
squaring  and  uniting  terms, 

(26-6>'  +  2a(26  -6>=a'  +  a'6«  -  2a'6, 


a;'-f-2ax: 


a»-{-a»6'^2a'6 
a' 


^.  +  2ax+a'=2^-^,, 


V26-6' ' 

whence,  x-=.  ±a  ■?  — : >  Ans, 

(235) 


TWO   UNKNOWN   QUANTITIES.  161 

83.  Squaring  the  left  hand  member  as  indicated, 
4  +  4x  +  x\    2b-\-tt  ^ 
4  —  4x  +  x^~      2b  "  ' 
clearing  of  fractions  and  uniting  terms, 

ex* — 4cx*  +  4cx=l  6bXf 

,     .        ,166 

or,  a:'— 4ar  +  4  =  — , 

c 

whence,  ar=2l  l±2y -I, -4n«, 

Note, — The  value  x=0,  satisfies  the  original  equation,  and  also  our  reduced 
equation  after  uniting  terms. 


EXAMPLES  OF  SIMULTANEOUS  EQUATIONS. 
(301,   page  243.) 

1.  From  the  second  equation,  we  have  x—2y*,  and  by  substitu- 
tion in  the  first, 

16y'-8y  +  l  =  121, 

4y=12  or  —10, 
y=   3  or  -   2n 
2y«=ar=18or      12^  j    ^'**' 

2.  The  second  equation  gives  by  transposing  y  and  multiplying 

ary=lly-|-; 

and  substituting  in  the  first  equation, 

3y'-f  22y=240, 
9y'  +  66y4-121=:841, 

3y=  — lldb29; 
"=     6or-13i,    ^^_ 


2a?=22— y;  x=     8  or      l7f 

(235-243) 


i\ 


162  QUADRATIC  EQUATIONS. 


3. 

By  substitution  from  the  second  equation, 
4y'  +  9y  =  100, 

2y  +  f=±V, 

and 

reducing, 

y=4  or  - 

-  61 

) 

9y 

,    >Ans. 

x=-f;                    x=9or- 

-14, 

M 

4.  By  adding  the  equations,  and  dividing  by  5, 

ar"  +  a;=12, 

x=       3  or  —  4  ) 
y=25— 5ar;  y=      10  or  —45  f  * 

5.  Multiplying  the  second  equation  by  3,  and  adding, 

13x'  =  52, 
a:'=4, 

x=±2)     . 

y=±3[^'"- 

6.  From  the  second  equation, 

y=40  —  4ar, 
a:y  =  40a;  — 4x'; 

hence,  the  first  gives  / 

,  a:'— |0a;=  — 336, 
ar*— ^0;c  +  400=     64, 

ar=+20±8, 
x=     28  or      12  )     . 
y=40-4ar;  ^=-72  or  -  8  j  ^'**- 

*l.  The  second  equation  gives  ^=-^  ;  whence, 

V=252, 
y'=36, 

y 

a?=fy;  x 

(243) 


±15  H'**- 


TWO  UNKNOWN   QUANTITIES.  163 

8.  From  the  second  equation,  x=^i/^ 

|ly»  +  4y'  =  181, 

181y'  =  181x25, 

9.  Adding  the  equations, 

ar'  +  2a;y4-y'=36, 

^  +  y=±6,  (1) 

•ubtracting  equations,  a:'— y'=  — 12,  (2) 

dividing  (2)  by  (1),  a:-y=  qF2,  (3) 

from  (1)  and  (3),  ""=  ^^  I  Ans 

10.  Adding  the  square  of  the  second  equation  to  the  first,  we 
have 

2a;' =  5, 


ft  n  AA/^      I     ^nS, 


11.  Put  af=vy; 


vy +  vy'=56,         or  3/'=  -5-—  ; 
'       v^  +  v 

v/  +  2y'=60,         ory'=--— ; 

whence  by  equating  the  values  of  y', 

v=s  <>*•  -I- 

With  v=|,  the  second  equation  gives, 

10y'=:180, 

±4*^2  [^'**- 
Withv=— J,  3y'=300; 

y=±io  I    . 

:r=^14H'*'- 
(243) 


164  QUADRATIC   EQUATIONS. 

12.  Let  y=vxy 

CO 

3x*-\-vx*=z  68,         or  a:'=- ; 

3  +  v 

160 

4i;V  +  3i;a;'  =  160,         or  a?"=T-i — S"  » 

whence,  68v"4-llv=120, 

or,  4i;'  +  Uv=Vt% 

,  .     11     .  /11\'     32761 

''  +17^+168/  =wr 

and,  i;=f  or  — f^. 

With  the  value  v=|,  the  first  equation  gives 

I7ar«  =  68x4, 
ar'  =  16;         x=±4      \ 
y==h5 

The  value  of  t> =  —  «^|,  gives  x=±:  — -  j  An*. 

16 


13.  Let  y=var; 

»  1+v 

rar"  — 2i;V=   1,         or  aj'  = — -, ; 

'  V  — 2v' 

whence,  24v*— llv=  — 1, 

i;=i  or  \. 

Hence,  if  v=|,  4x'=:36,  a;=±3 

y=±i 

If  v=|,  9x'=96 

(243) 


TWO   UNKNOWN   QUANTITIES.  165 


14.  Put  y—vx\ 

X*—  va:«-fvV=     21, 

vV  — 2var'=  — 15, 
whence, 

21 

,       -15 
or  x  =-= — —  ; 
v'— 2v' 

36y'-67t;=  — 15, 

v=     f  or  1. 
From  the  second  equation, 

jf^=i,  /-6y'=-15;  y=±Vd    ^  ^"*- 

ar=±34/3 

15.  Add  1  to  the  first  equation,  and  take  the  square  root,  then 

ar  +  y=i:  +  10  or  —12, 
xy—y*=     8. 
Taking  ir  +  y=  +10,  the  second  equation  gives,  by  substitution, 

y'_5y=_4; 
whence,  y=4  or  1  )     . 

x=6  or  9  J 
Taking  ar  +  y=  — 12,  we  have 

y«  +  6y=-4j 
whence,  y=  — 3±V^5  )     . 

ar=-9::pf^5  i" 

16.  Put  y=var;  then 

6x*  +  2v*x*  -  5vx* =12,        or  «'  =         ^  ^ 


3vV— 3a;"— 2vx'=   3,        orx'  =  — r 


6  +  21;"— 6v' 
3 


Hence,  lOv'— 3v=18, 

t;=i  or  -I, 
If  we  take  v=f,  the  first  equation  gives 
3x'  =  12, 

(244) 


whence. 


166  QUADRATIC   EQUATIONS. 


If  t;=- 

-|,  we  have 

372x«=r300, 

,_300_3*4-25  ^ 
*"~372     3- 4 -31' 

wbence, 

.     5     ) 
V31  1 

1 

y  ^n*. 

1 

IV.  Multiply  the  second  equation  by  2,  and  add  the  product  to 
the  first;  then 

a:«  +  2xy  +  /=  121, 
subtract  it ;  a:'  —  2xy  +  y'  =  9, 
whence,  ar  +  y  ==fcll, 

a?— y  =db  3;  whence,  a:  ==t 7    or  db4  ) 
y=db4    or  ±7  ) 


18.  Squaring  the  second  equation,  and  subtracting  the  square 
from  the  first,  we  have  2ary  =  80  ;  adding  this  to  the  first,  and  taking 
the  square  root, 

ar  +  y=±13, 

K^y=-       3  ;  whence,  a:=4-8     or  —5  )    . 
y=  +  6     or  — 8  ) 


19.  Dividing  the  first  equation  by  the  second, 

a;'-  xy  +  y^=  273, 

square  of  second,  a;'  +  2ry  +  y'=  324,  (1) 

3.ry         =      61, 
xy        =      17. 

Subtract  4a:y  from  (1), 

a:'— 2ary  +  y'=   256, 
hence,  ar-y  =  =fcl6, 

a:4-y=     18;  whence,  ar=  17  or    1  \a 
y=   1  or  17  3 
(244) 


TWO   UNKNOWN   QUANTlTiESr^^ss^s:^^^^''^     167 

20.  Adding  three  times  the  second  equation  to  the  first, 

a:'  +  3a;V  +  3x/  +  y'= 729, 

taking  the  cube  root,  x  +  y=     9.  (1) 

The  second  equation  gives,       xy(x-\-y)  =  180'y  (2) 

dividing  (2)  by  (1),  a-y=   20.  (S)  -i 

Subtracting  four  times  (3)  from  the  square  of  (1), 

ar«_2xy  +  y'=     1, 
or,  x—y=±l,  (4) 

From  (1)  and  (4)  we  have  x=5  or  4  )    .  ^ 

and  y=4  or  5  ) 

21.  Performing  the  multiplications  indicated,  we  have 

x'--x'y+   xf-f=    13,  (1) 

ar'y-  xy^         =6;  (2) 
(1)— twice  (2),           a:'— 3a;'y  +  3ary'— y'=     1, 

taking  cube  root,                                 ar— y  =     1,  (3) 

from  (2)  divided  by  (3),                         xy  —     6,  (4) 
(3)  plus  four  times  (4),             a;'  +  2a:y  +  y'=   25, 

a:  +  y=±5;  j(o) 

hence,  from  (3)  and  (5),                                          a;=3  or  —2  )    . 

y=2  or  —3  J 

22.  Dividing  the  first  equation  by  a;  +  y  and  transposing,  we  have 

ar'  — 2a;y  +  y'  =  0, 

ar-y  =0, 

second  equation  is  ar  +  y  =4  ;  a;=2  ) 

y=2) 


Arts, 


23.  The  first  equation  is  a  quadratic  in  xy ;  and  completing  th< 
square,  we  have 

56i:y-24a?y4-4=    16, 
6a:y— 2  =  =fc4, 

xy=     l,or-i, 
and  since  ar=2y,  2y'=     1  or  — | ;       y—  rhi|/2  or  =t^ 

ar=db  ♦/2orzbif^. 

Ans, 
(244) 


168  QUADRATIC   EQUATIONS. 

24.  Dividing  the  first  equation  by  the  second,  we  have 

squaring  the  second,  ir'  +  2 ry  +  y'  =  1 44  ;  (l) 

hence,  3ary        ==144  — |ry, 

or,  a-y         =   32,  (2) 

(1)  —  four  times  (2),  a:'  — 2i:y  +  y'=    16, 

whence,  a;— y=:d=4, 

.+y=   12;         .=8  or  4) 
y=:4  or  8  ) 


25.  We  make  use  of  formula  (C),  page  242,  in  which 

*=8,     5'  =  64,     «*  =  4096. 

2p'— 256;?  +  4096  =      2402, 

p'-\2Sp=—   847, 

y— 128;>  +  4096=      3249, 

jr>-64  =  rfc      67, 

xy=:pz=.        121     or  7. 

Taking  a-y=7,  and  combining  it  with  x4-y=8,  we  get 

.-y=±6;  x=7     orl)     ^^_ 

y=l     or  7  ) 


If  iry=121,     ar— y  =  rbi^— 420  =  ±2*^— 105  ; 
whence,  ir=4rfcV-105  "I  ^^^ 

y=4::^i^-105  J 

26.  Dividing  the  first  equation  by  the  second,  we  have 

z'*-{-x_x(x-\-\)_2 
jr'  +  l        a:'  +  l  "~3* 
X  2 


or, 


x'—x+l     3' 
whence,  2x*  —  52r  =  —  2, 


.=     J±j;         x=2     or    |)^^^_ 
y=2     or  16  ) 
(244) 


TWO   UNKNOWN   QUANTITIES.  169 


27.  Dividing 

the  first  equation  by  .r+y. 

x*—  xj/  +  y^  =  2xi/y 

x'-2;ry  +  y'=:a:y-16; 

a--y  =  ±   4, 

x'  +  2xi/  +  y'=      80, 

x-\-yz=dz4y5; 

whence, 

x=db2V5±2 
y=dz2V5zp2 

28.  Put 

J= 

p, 

y*  =  Q  ;  then  the  equations  become, 

•whence, 

P+^  =  f^a  +  6,                           (] 

and 

P'-Q'=a-b, 

and 

--^'-^^       (^ 

by  (1)  and 

(2). 

2P=^+Va  +  b,  ' 

Ant. 


Va  +  b 
2«  =  '^*+.S' 


and  wo  obtain  finally,  P=^ .        or  -=(j~)\ 


29.  Dividing  the  second  equation  by  the  first,  we  have 

x-y-l-     ^' 

or,  (ar-y)«— 4(ar-y)  =  -4, 

(^-y)*-4(ar-y)  +  4=     0, 

hence,  x—y=     2. 

Hence  the  second  gives  a;  +  y=     8; 

x=z5  ) 

y=3  3 


Ans, 
y=3) 

(244-^245) 


170  QUADRATIC   EQUATIONS. 

8^.  Adding  and  subtracting  twice  the  second  equation, 

ar«— 2jry  +  y'=a  — 26; 


x-y=±Va--2b; 

^  , '  , >  Ans, 

y  =  rfcif^a  +  26^lVa--26  J 

il.  We  have  x=—  ;  whence, 

y 

y*  +  2ay'=46', 
y*4-2ay*  +  a'=a'  +  4J'; 

whence,  y=:i={  -a±f^o'  +  46«  }^  ) 

in  the  same  way,  x=  dz  |     a±^a'  +  46'  [  ^  ) 

32.  Squaring  the  first  equation,  we  have 

^'>  Vr^y%  V^)-  (-f  fsj  y'.4-_y  =  333,  (1) 

a)ividmg  (1)  by  (2),  we  have  a  quadratic  in  a:y, 

'  -  -^  ^v^eTc^  xy=   36  or  3^/^^  ^  '   ^^  ^'   ' 

/Substituting  a?y=36  in  (1),  37y=333 ;  y=     9 

;l4>J^_/."  ^y=3Vi«(l)»  37y=333-36;     y=324 

33.  Adding  the  equations. 


^»*. 


x-\-2Vxy  +  y=a-^h^ 

Vx+Vy^V^^'^  (1) 

(245) 


TWO  UNKNOWN   QUANTITIES.  171 


subtracting  them  x—y=a—h, 

Hence,  from  (1)  and  (2),  x— 


''-''=^.-       (^) 


a-{-b 


34.  Put  a:^=P,     y'  =  §,     then 

P  +  ^  =  6. 

The  solution  is  the  same  as  that  of  19. 


35.  Clearing  of  fractions,  the  second  equation  gives 

the  first  is,  x-\-y=lO\ 

whence,  a:y  =  16. 

Combining  the  two  last  equations,  x=8  or  2  )  ^ 

y=2  or  8  j 


36.  Dividing  the  second  equation  by  the  first,    v  ^-^y  ^  -  >9>  ""  -  4y   ^ 

whence,  a;*4-y^  =  ±8,  (1) 

a;— V=    16, 

by  di vision,  x^  -fy*  =  ±  2 .  (2) 

Taking  the  positive  signs  in  (0  and  (2),  we  have 

x^=/b,        y*=     3; 
negative  signs,  a:*=— 5,         y*  =  — 3;  x=:25  \    . 


yzx:    9 

(246) 


\ 


172  QUADRATIC   EQUATIONS. 

37.  By  transposition,  and  squaring  the  second  equation,  we  have 

a.  3 

a. 
and  substituting  this  value  of  y^  in  the  first  equation, 

3. 

x-Vx=2, 

♦/j-  =  2or— I;         a;=4orl 
From  the  second  equation,  y=S 


>  Ans. 


38.  Adding  twice  the  second  equation  to  the  first,  we  have 
(i:Vy*)'  +  2(a;^  +  y^)  =  35, 


±       1 
From  this  quadratic  in  ar'  +y^, 


a;*+y'i=  —  l±6,  =5  or  ~7; 


combining  this  with  the  value  of    a;^y', 


x'-y=  =  ±l,  or  ±5; 


whence,  r  x=2l,    8,-1,-216)    ^ 

.^~-fi^^.^r~^r^-/    y=   8,27,-216,  -l[^^'- 

1  1 

39.  Put  Pzzzx^^  ^=y*,  and  the  equations  become, 

P'+Q'  +  P+Q  =  2e,  (!) 

PQ=   8.  (2) 

Adding  twice  the  second  to  the  first, 

(^+cr+(^+«=42, 

P+^==6or-7,  (3) 

P'-2P^+  ^'  =  4  or  17,      _ 

P-Q=^±2  or  rtV^17;    (4) 

hence  from  (3)  and  (4),  P=^=»  =     4,  2  or  {  {-1±^), 

■  Q=y^=     2,4ori(-7±m); 

whence,  x=±%,  ±2V2  or  ±\^(-1±y\^)  }H  ^^ 

y=     32,1024or       S^( -7:pVl7)  !  *   ) 
(245-246) 


TWO   UNKNOWN   QUANTITIES.  173 


40.  The  first  equation  is  a  quadratic  in  -;  and  adding  4  to  each 


side,  and  taking  the  square  root, 

V^_3        _11 
fV-2  ""^       2  ' 
4a:=:9y  or  121y. 
From  the  second  equation. 


ar=9  or  f ff  ) 
y=:4or^VTi 


Ans, 


41.  Here  P=x^,  g=y*    '  ^^^  ^  ^  ~ ^ 


P'Q'=2Q\  -yi^\^           (1) 

SP-Q=U;  V^-^             (2) 

from(l),                               P'=2(2,  rl~^      xK//^   (3) 

"     (3)  and  (2),        P'--16P=-28,  ^  J^^  _/^^~^'^^  ^ 

hence,  a:^=P=8±6  =  14  or  2, 

y^-Q—  98  or  2; 

whence,  ic=2744  or  8  ) 

y=9604or4[  ^^*- 


42.  LetP=ar*,  §=y*;  then 

P'  +  PC+C'=     1009  =fl,  (1) 

P*  +  P'^'+^*  =  582193  =  57Ya, 
Add  7^'  C',and  take  the  root,  P'  +  ^'  =  VsTT^+P'V  (2) 

subtract  (1)  from  (2),  a-- P q-VWu^^'q\ 

a*-2aPQ-\-P'Q^=5l1a  +  P^Q\ 
2PQ=a-5l7, 
PQ  =  216,  (3) 

Adding  (3)  to  (1)  and  )  r>    '0-4... 

taking  the  root,  )  ^+V-±35, 

subtracting  three  times  )  __ 

(3)  from  (1).  )  /'-«=±19; 

(246) 


174  QUADRATIC   EQUATIONS. 

^     .jhence,  /  /  «*  =:P—       8  or  —27, 

^^  Y'*  -  :i).  -^ ,  <r^  -^y  A3.^-±Sr:^l9y  =^=-27  or       8, 
^       whence,  Wt:i=/C,i-^f,  Li-^.  -/6     ^=81  or  16  ) 

43.  Complete  the  wjuare  of  eacn  equation  Dy/adlaing  16  ar  to  t 


first,  and  x  to  the  second ;  then 

y— 2ar^y'+   ar=       4+ar; 
taking  the  roots,  y— 4f/a;=±4V4+^,  (1) 

i^y-Vx=±Vl+i;  (2) 

(1)  minus  four  limes  (2),     y— 4V'y=     0  ; 

i^rhence,  y=16    )  ^^^^ 

x=   2|  ) 

44.  These  equations  may  be  written, 

ary(ar  +  y)=   30,  (1) 

— ^=     -.  (2) 

xy  6  ^^^ 

Dividing  (1)  by  (2),  xy=  36  ; 

^ac^ /'/^v ^^^y  -  ^V  ,  y  -^  y  ^yv      ^y=±6,  (3) 

^from  (2)W^,        /  (x  +  y=±:5, 

'  V/  ^^  v^       r^  ^'  -^'^  ^'  '  ^  /*=3»  2,  1,  or  -6  )  ^ 

/  '        '       V        ^ — — -^ 

'  ~'   .  -?•  7  '4^  Dividing  the  first  equation  by  the  second,  we  have 

^V=   16, 
a'y  =  it:4. 
Hence,  a^4-2ary+A'=   16  or     0, 

a:"— 2xy+y'=     0  or    16  ; 
ar  +  y  =  ±4  or      0, 
x—y=.     0  or  ±4  ; 
whence,  ^=±2)    ^^    i^=±2) 

y=±2j  (y=:F2) 

(246) 


TWO   UNKNOWN   QUANTITIES.  175 

46.  Dividing  the  first  equation  by  the  second, 

x*  +  x'y+xY-hx/  +  y'  =  lOSl,  (1) 

(x-y*)-x*^4z'yi-6xY^4xi/'-\-y*=     81 ;  (2) 

(1)  minus  (2),  5x'i/—5xY  +  5xi/*=  960, 

dividing  by  5ary,  x*—xy-\-y*—  , 

xy 

(ar-y)*=a:»-2xy  +  y'=       9,  (3) 

ary  +  9iry=   190, 

ary=     10  or  —19;     (4) 

from  (3)  and  (4),  a^+y=±   V, 

"whence,  ar=5     or  —2  )    ^ 

^  ^\  Ans, 

y=2     or  —5  ) 

47.  We  have  given 

x*+xy-\-y^=       7,  (1) 

x*+xy  +  y*=  133.  (2) 
Adding  xY  to  (2),           x* +2xY  +  y*  =   133+  xYi 

or,  a:'  +  y»  =V^133+a:y.  (3) 

Taking  (1)  from  (3),  1 -xy=yU3-hxY, 

49-Uxy  +  x'y^=   133+a:y. 

xy=   -6.  (4) 

Combining  (1)  and  (4),  ar+y=    ±1, 

also  •  x—y=   ±5. 

The  values  o(  x-\-y  and  x—y^  may  be  combined  in  four  ways,  as 
follows : 

x  +  y=-{-l;     x  +  y=z-^l]     x  +  y=-\-l;     x  +  y=  —  l; 
ar— y=4-5;     x— y=     5;     x—y=—5;     x—y=  —  5; 

^.  .  j        a;=+3;  a;=+2;  ic=-2;  a;=-3; 

Wivmg       I        y=-2;  y=-3;  y=  +  3;  y=+2. 

Or,  condensing  the  expression, 

y=T3    or  ±2  ) 
(246) 


1 


176  QUADRATIC    EQUATIONS. 

48.  Eliminating  y  from  the  last  two  equations,  we  have 
52^'  — 36^=— 5, 

whence,  x—\     or  ^^ 

y=-y-;  y=i     or  f| )  ^n*. 

z=i     orH 


PROBLEMS  PRODUCING  QUADRATIC  EQUATIONS. 
(314,  page  258.) 

1.  Let  ar=  the  greater  part;  then  14— ar=  the  less.     Hence, 
Oar        16(14-a:) 


14-a:  X 

9ar'  =  16(14~a:)', 
3ar=4(14-ar), 


2.  Let  a:=  the  number  of  persons  ;  then 

350  350 

+  20  = 


8  and  6,  An%, 


X  x—2 

Hence,  ar*— 2x=35, 

x'— 2a;  4- 1=36, 

a;— 1  =  6;  x—*J,Ans» 

3.  Let  ar=  the  number ;  then 

(22-ar)ar=     117, 

or,  x'  — 22ar=— 117, 

ar"-22ar+121=         4, 

ar=ll±2;  ar=13  or9,  v4n«. 

(246-259) 


PKOBLEMS   PRODUCING   QUADRATICS.  177 

4.  Let  «=  one  part,  and  18— ar=:  the  other;  then 

X*  25 

X         5 

18— a:"~i* 

9.r=90;  10  and  8, -4w«. 

5.  Let  x=  one  number,  and  ar— 4=  the  other;  then 

(2a;— 4)(a;-2)8  =  1600, 
x*  —  ^x=     96, 

ar=2=bl0  ;       12  and  8,  Am, 

6.  Let  x=z  one  number,  and  y=  the  other,  the  less ;  then 

^^^  x-y  :  y  ::  4  :  3 

^-^^r/^5^  Sar^Ty,  (1) 

^^^,    V^--^^  ary'=504,  (2) 

y"=216;  whence        y=  6)    ^ 
ar=14  J 

7.  Take  8a;  and  6a;  respectively,  for  the  length  and  breadth  of  the 
field ;  then 

8a;x5a;  a;» 

— -— — — r=  the  number  of  acres  =— . 

loO  4 

Hence,  -  x  8«rl3  x  26a;, 

a;'=13xl3, 
a;=13;  8a;=  104,  length    1 

6a;=  65,  breadth  [  ^'**' 

8.  Take  5a;  and  4a;  for  the  length  and  breadth  of  the  stack ;  \» 
will  be  the  height.  Hence  there  are  70a;"  cubic  feet  in  the  stack, 
and  20a;'  square  feet  on  the  bottom.     Therefore, 

70a;*x4a;=224x20x», 

a;'=16, 

a;=4, 

Length  =5a;=20;  Breadth  =4a;=16;  Height  =Ja;=14,  Ans, 

(269) 


178  QUADRATIC   EQUATIONS. 

9.  Let  «'— 7  be  the  number;  then  from  the  statement  of  the 
problem,  we  have 

or,  a:'  +  9=81  — 18a:  +  ar", 

x=4  ;  ic'  — 7=9,  Ans, 

18 

10.  Leta:=A's  number  of  eggs;  then  — — =A*8  price  per  egg; 

100— a:=B's         "  "     ;  ~        =B'8       "         « 

X 

Hence,  since  they  received  the  same  amount  of  money,  if  we 
multiply  each  man's  number  of  eggs  by  his  price  per  egg^  we  have 
the  equation, 

183r     _  8(1 00 -a:)  ^ 

100-ar~         X         ' 

or,  9ar'=4(100— a:)', 

3ar=2(100~-a;), 
a?=40;  A,  40;  B,  60,  ^n*. 

11.  Here  we  have  from  the  statement, 

a;  +  y=   6, 
x«  +  /=72, 
«ind  the  solution  is  like  that  of  19,  (301). 

no 

12.  Let  x=z  the  number  of  miles  per  hour ;  then  — =  the  num- 

X 

)>eT  of  hours.     Whence  from  the  conditions  given,  we  have 
36       36 

or,  a:'  +  ar=12, 

x*+x  +  \  =  *^;  xz=z3^An8. 

13.  Let  X  and  y  be  the  numbers ;  we  have  from  the  statement, 

ar  +  y=100,  (1) 

Vx-yy=2; 
squaring,  a;— 2V^+y=4,  (2) 

from  (1)  and  (2),  _Va:y=48,  (3) 

four  times  (3)  +  (2),  ar + 2 Viy  +  y = 1 9  6, 

Vx+Vy=l4, 

Vx=8\     ar=64,y=36,  ^w». 
(259-260) 


PROBLEMS  PRODUCING   QUADRATICS.  179 

675 
14.  Let  a?=  the  number  of  pieces;  then =  cost  of  one  piece, 

X 

and  from  the  conditions  given, 

48x=675+— , 

X 

48a;'— 6'75a:=675, 
or,  16j;'  — 225a;=:225;  x—\b^Ans, 

See  solution  of  Exanaple  3,  Art  (J1805), 


15.  Let  x=z  the  price  of  cloth  ;  then  rT:r=  gain  per  cent.,  and 

a:' 

— -=  the  whole  gain.     Whence, 

a;»4-100ar=3900, 

a:=— 50±80  ;       a;=$30,  Ans, 


4x      104a; 
16.  Let  x=  the  purchase  money ;  then  a;  +  ---^=  =the  whole 

104a; 
cost,  and  390 — 77^=  the  gain.     By  the  statement  we  have  also 

the  gaiu  =  ^^.  -•  — .     Hence,  i^  -  ^^^  ,  .  1  . 

^'/^    ^/-'  ^:j^  '  •  ti^* ' 

104.r       104a:'        ^^  -^A^Z- 

or,  / /~/f-t^  ^  ^  ^  =z^^^^ 

2j;»  -.  -^^^ 

3000-8a;= .         )L^^/ T-CHJ  \l  ~^/^^^;>^<5^ 

300  r-  »^ 


Put  a =300,  and  divide  by  2  ; 


a;' 
6a  — 4a;=— , 
a 


a:'  +  4aa:  +  4a'  =  9a', 

a-— a  ;  a:  =  |300,  .^tj*. 

(260) 


180  QUADRATIC  EQUATIONS. 

17.  Observe  that  396  — 216  =  180  miles,  B's  distance.     Let  x-=. 
the  number  of  days  they  traveled  ;    then =  A*s  rate,    and 

=  B's  rate.    Hence, 

X 

_216_180 

~   X  X    * 

a:«=36, 
ar  =  6  ;  A,  36  ;  B,  30,  Ans, 

11*.  'Vrith  two  unknown  quantities,  a:+y=60,    «y=704 ;    Seo 
(15J99),  Example  3.     With  one  unknown  quantity, 

60.r— x'  =  '704, 
ar'—60jr  + 900  =  196  ;  x=^4A  or  16,  Ans, 

19.  Let  xz=  price  of  sherry  per  dozen, 
y=      "         claret 

Then  1x  +  l2i/=50; 

10     6 
—=-  +  3, 
X     y 

lOy 
or,  x^ 


^  3y  +  6 

By  substitution^      VOy  +  36y'4-'72y= loOy  +  300, 

9y'  — 2y=V5;  y =Sy  x=2y  Ans, 

20.  Let  19ar=  the  distance  from  C  to  D ; 
then,  x=  B*s  rate  per  day,  and  also  his  number  of  days, 

ar'=  B's  distance, 
'7x4-32=  A's  distance. 

Hence  we  have, 

ar'  +  7a;  +  32  =  19ar, 
ar'  — 12a;=— 32, 
a;=6±2, 

a;=8or4;         ldx=zl52  or  I6y  Ans, 
(260-261) 


PROBLEMS   PRODUCING  QUADRATICS.  181" 

21.  Let  x^=  the  bushels  of  wheat, 
a;+16=        "         "         barley. 

Hence, 


24_ 

X 

=    24        1 
a;+16     4* 

(25 

cents 

=iofa 

doUa 

24a? +  16-24  = 

=24.+^'-^;«^ 

1 

.  -'*'" 

=  384, 

^'  +  4^^+16  = 

=  400, 

X 

2~ 

=  -4±20; 

32, 

and  48, 

Ans, 

22.  Let  2ar=  the  distance  from  C  to  D, 

.,    ,.  4(x— 18)      .- 

ar  +  18=  As  distance,  — •'=A8  rate  per  day ; 

DO 

ar-18=  B's        "  -'^  =  B's     " 

28 

Tlie  distance  A  traveled  divided  by  his  rate  per  day  gives  the 
number  of  days  he  traveled  ;  and  since  B  traveled  the  same  number 
of  days,  we  have  the  equation, 

63(2;+18)_28(a;-18) 
4(j;-18)~     ar-flS     ' 

or, 

9(ar  +  18)'=16(ar-18)', 

3(ar  +  18)  =   4(a;-18), 

ar  =126  ;  2jr=252,  Ant, 


23.  Let  «=  one  number;  y=  the  other;  then, 

(;c-y)(x'-/)=  32. 
{x  +  y){x'+y')=2l2\ 
or  multiplying, 

«•— ary'— ar'y  +  y'=   32, 
«'  +  ary'+a;V4-y'=272. 
(261) 


182  QUADRATIC   EQUATIONS. 

Adding  and  subtracting  these  equations  and  dividing  by  2, 

a:'  +  y'  =  152,  (1) 

X2/'+x'y  =  l20,  (2) 

(l)  +  3-(2),  x*-hSx'y  +  Sxy'  +  y'  =  512, 

taking  cube  root,  x-\-y=     8. 

Dividing  the  first  equation  by  this,  and  taking  the  square  root, 

a:— y=±2;  ir=5,     y=Z,  Ans, 


24.     Let  x=z  the  number  of  horses  B  put  in, 

— =  price  of  one  horse  a  week,       X  -  h»   i  ,  y-  .J)e 

4-18    *  Y-it;/f  .•■•^:y 

1-18==  rent  of  pasture ;      Y~^  ^0  :  2-0  :  4  ,' y  -^  2- 

when  ar  becomes  a: +  2,  yyy.^^.  T^Oy^/Zfi 

-=  price  of  one  horse  a  week.    *L^-2.  y^  -  //  p- 

4-20  Y^y^2.// 

^-j-^+20=  rent  of  pasture.      y^^2_^y  ^/fy  =z  /iZ- 
Therefore,  y-^^  &y  =:.  ^?_ 

36       40 


X      x-\-2 
x*  +  6x=l2y 

x=   5.  Rent  =30  shillings,  Ant, 


26.  Let  «=  the  figure  of  the  ten's  place, 
v=         "  "         unit's   "    . 


I0x  +  y_     ^ 

— Z7.      —        » 

(1) 

xy 

I0x+y  +  2l=     lOy+ar; 

(2) 

9z-9y=-2l, 

x-y--  3, 

x=     jf-3. 

(261) 

PROBLEMS  PRODUCING  QUADRATICS.         183 

Substituting  in  the  first  equation,  we  have 
2y'-.l7y=-30, 

X—       3 ;  36,  Ans, 

26.  Let  xr=:  the  first  number,  which  is  the  least ;  y=  the  second, 
and  2=  the  third.     Then 

y^/^oV,    '^  a:-27+z=-   6;  (1) 

'  ,/  y  a:  +  y  +  0=      33;  (2) 

:  ^    /  ;r'  +  /  +  z'=   441.  (3) 

^  '  ^  ^From  (1)  and  (2),  3y=     39, 

y=     13; 

substituting  in  (1),  x-\-z—     20, 

"  (3),  a:'  +  2r»=    272; 

a;»  + 22:2  +  2'=   400, 

2x2=   128, 

a;— 2=±12, 

x=       4, 

2=     16  ;         4, 13  and  16,  An9, 


27.  Let  x  and  y  be  the  numbers ;  then 

«y=   24,  (1) 

«+y+a;'+y'=  62.  (2) 

Adding  twice  the  first  to  the  second, 

(•^+y)'+(^+y)=iio, 

ar  +  y=   10, 

x—y—     2 ;         *=6,    y=4,  ^w*. 

28.  Let  «+y=  one  number,  and  x—y=:  the  other;  then 

«»-y'+2ar=     47,  (1) 

2x"  +  2y'  — 2a:=      62,  (2) 

4a;'  +  2ar=   156, 

«=       6, 

y=       1 ;  7  and  4,  Ans, 

(261-262) 


184  QUADRATIC   EQUATIONS. 

29.  Let  x-=.  one  number,  and  y=  the  other;  then 

ar+yz=   27, 
a:'  +  y'  =  5103. 

Solution  the  same  as  that  of  Example  19,  page  244. 

30.  If  ar=  one  number,  and  y=  the  other ;  then 

x^y  =        9,  (1) 

ar*  +  y*=24l7.  (2) 

See  Example  1,  (301). 

31.  Let  x  +  y=  one  number,  and  a;— y=  the  other;  then 

2(x«-y')(jr'  +  y')  =  1248,  (1) 

4xy=     20.  (2) 

From  (1),  x*—y*=:  624, 

eliminating  y,  ar"— -624x*=  625, 

x*=  625, 
rr=  6, 
y  =       1  ;  6  and  4,  Ans. 

32.  Let  ar=  the  number  of  days  it  takes  one  man, 

a;  +  10=         "  "  "  the  other ;  then 

12  .      12 


z     //...    ^  /7  0  ^»-14ar=120. 


y  ^-^  /<.  y.    ^/T-O 


ar=7±13 ;         20  and  30,  Ans. 


33.  Let  ir=:A's  stock, 

1000-a;=B'8     " 
1140— ar=A's  gain, 
640— (1000-*)=  X  —  360  =B'8     " 

Now  A's  stock  must  be  to  B's  stock,  as  A's  monthly  gain  to  B*8 
monthly  gain.     That  is. 


>Ch  -7  ^fO-O-O  ,^^^  1140— a;      ar— 360 

-f    i 


;>V;/'U^y~>7--^   ^'^^^^^^^^^^ 


PROBLEMS   PRODUCING   QUADRATICS.  185 

Or,  by  taking  the  products  of  extremes  and  means, 

(3?— 360).r_(1140-.r)(1000-ar)  ^ 
_  _  _  ; 

whence,        d{x^—360x)z=2{U40000-2U0x+x*), 
a;'  +  3200a:= 2280000, 
ar  + 1600  =  2200, 
ar=600; 

A's,  $600  ;  B's,  ^400,  Ans. 

34,  Let  2x*=  the  number  in  the  first  drove, 

4  +  4t=  ''  second     " 

6ar'  +  12ar+12=  "  third 

3x' 4-   6^  +  16=  "  fourth      " 

Hence,  we  have 

lla:»  +  22:c  +  32=:1121, 
or,  a:'  +  2a:=:99, 

x=9y 
2.r'  =  162; 
therefore,  1st,  162  ;  2d,  40  ;  3d,  606  ;  4th,  313,  Ans, 

35.  Let  x=z  one  number,  and  y=  the  other ;  then 

Sxy-x'-y'  =  U,  (1) 

2ary-a:'  +  y'=14.  (2) 

Assume  vx=y ;  then 

11 


Svx* — a;'  —  v*x*  —  1 1 

X*-                  • 

\j  Vim^              •*/               t/    »*•     -— ■  A  A  * 

3t;-l-v" 

2vx*-x^  +  v*x^=U^ 
By  equating  values  of  a:',  we  obtain 

x«-       ^^ 

2v-l+v«* 

25v'— 20v=- 

-3, 

If  V=:i, 

whence. 

f 
25; 

ori. 

5  and 

36.  If  x=  one  number,  and,y=  the  other ;  then 

x-hy=    20,  (1) 

a:y  =  9216,  (2) 

xy  =     96, 
«— y  =       4  ;  whence,       12  and  8,  Ans, 
(262) 


186  QUADRATIC    EQUATIONS. 


37.  Let 

X  aud 

y  be  the  parts  ;  then 

xY  =  h, 
xy=Vh, 

(1) 

(2) 

whence, 

«-y=:(a'-4V6)^; 

)  Am, 

This  is  a  general  solution  of  Example  36. 


38.  Let  X  and  y  be  the  numbers ;  then 

:r=aV,  (1) 

xy=h\  (2) 

By  division, 

y"=-i ;     or  y=-,  «=a5,  Ans, 
a  a 


39.  Let  X  be  the  first  of  the  consecutive  numbers ;  the  number 
sought  will  be  ar(a:-fl)(2r  + 2).     Hence, 

(x  +  l){x  +  2)  +  x{z+2)-hx{x-hl)  =  lA 
or,      (y-  f)  %(j^^f]  ar'  +  2:r=  24, 

.  )fVy  f  y'^/-^  ^V>^  =7^  a:=4or  -6; 

^u'^- Vrwhence,  4- 5-6=     120  | 

^  .  y-    or,  (_6)-(-5)-(-4)  =  -120  [^~*- 

40.  Let  a:=  width  of  the  engraving  ;  and  2x=  the  length; 
lien  2x^=  square  contents  of  the  engraving, 

X^  C  18x4-4x3'=;  "  "        margin.     Hence, 

->^-f^    _^  2a;'— 36  =  18a; +  36, 

U^^77^-hI(;^  ^/.  ^*-3'^^^^^-9a;=36, 

Ly-^    _jwheDce,  a; =12,  -4w*. 

If^^^i^  :^  ^^'^C  (263) 


PROBLEMS   PRODUCING   QUADRATICS. 


187 


41.  In  order  that  the  two  lots  may  be  embraced  in  a  single  inclos- 
ure  of  six  sides,  they  must  be  placed  as  in  the  following  diagram. 

Let  ir  =  a  side  of  the 
greater  square,  and  y  a  side 
of  the  smaller.  Then  x*-\-y* 
will  be  the  area  of  the  two 
lots,  and  3(a?-f-y)  +  (ar— y)  or 
4a;-f-2y,  will  be  the  length  of 
the  fence  required  to  inclose 
them.     Hence 

a;*  +  y'=:4100, 
4ar  +  2y=280; 
from  (2),  y=140— 2:r, 

substituting  in  (1),    .a;'  +  (140--2ar)'=4100, 

a;'— 112a;=— 3100, 
a;-56=±6, 

ar=:62  or  60 
y=16  or  20  3 


0) 
(2) 


An9, 


42.  Let    rr=  the  first  portion, 

a— a;=  the  second  portion,  putting  a=1300, 
y=  the  first  rate  of  interest, 
r=  the  second         " 
Then 

xy={a-x)z, 
arz  =  36, 
(a— a?)y=49. 
We  have 


(1) 
(2) 
(3) 


49 


V— 


36 


and  eliminating  y  and  z  from  the  first  equation, 
49a:"=36(a-ar)', 
a?=:600; 
whenoe,  y=7  per  cent.,    ^=6  per  cent.,  Am* 

(263) 


/-  r  I  ^y 


I   V  I 


^xL 


ts 


r't.e/  188 


QUADRATIC   EQUATI0N8/^^^^^W'<^/-       v 


•L    2  r         '*-?•  ■'^^^  ^~  ^'*  number  of  hours,      y^  /  J"  ^  ,'/  2,  /  y 

»i=  the  distance  from  London  to  York; 

then  *  2i>^f,^f^    ^  y:     -h^^y 

I  —  =  A's  miles  per  hour :     —  =  B's  miles  per  hour;   ^ 

>/  *  ,/  y  ^^^Qc^^-^c^ 

^'^  ^^^.u:^:;  ^L>^Z::)  "^^^^^^^n  ^^  z!n  -  ^-  /^^ 


a;       y 
ar— 25=y— 36, 


or,  ^  _+_=l,  (1) 

and  ar-25=y-36,  (2) 

since  they  traveled  the  same  number  of  hours  before  they  met. 

Hence,  from  (2),  a:=y  — 11, 

and  from  (1),  y'  — 72y=  — 396  ; 

therefore,  y=66,         ar=55,  Am, 


Street. 


44.  Let  ar=  the  side  of 
B's  lot. 

From  the  conditions  we    6 
have 

36  +  6jr=(ar— 6)ar, 
a;'  +  12jr=3a, 

x=6  +  6V'2; 
whence,  6(1  +♦^2),  Ans, 


6 

X 

A'a  lot 

B'3  lot. 

45.  Let  ar,  y  and  ^  be  the  three  numerical  quantities.     Then 


Hence, 


y'+«'+y+2=50. 

«'-y«  +  2r-y=10, 

22' +  22  =  60, 

2'  +  z  =  30, 

2=5  or  —6  \ 
y=4  or  —5  V  -4nj. 
a;=3  or  —4  ) 
(263-264) 


(1) 
(2) 
(3) 


PROBLEMS  PRODUCING  QUADRATICS.         189 

46.         Let  x-=.  the  side  of  the  cube  ;  then 

a:'=  the  number  of  solid  units,    /x^7''  ^  ^'^    d^A^-  ~i 
V^=.  the  diagonal.  t^'ly^Y'^  -  ^<~^-  ~  i/^y^ 


Therefore,                                 x^^Vzx'', 

1 

x«  =  3x'; 

whence. 

x=X/Z,  Am. 

47.  Let  X  and  y  be  the  numbers  ; 

x-\-y=xy\ 

from  (2)  and  (3),                                  iry=3, 
whence,                                            x—y=±.^—Z 

(1) 
(2) 
(3) 

and,                                     x=i(3±i^-3),  y=i 

(SrpVCrs),  ^n5, 

48.  Let  X  and  y  be  the  numbers  ; 

x-^y=zxy,  (1) 

a:«-y»=ary;  (2) 

whence,  x—y—\.  (3) 

Subtracting  the  square  of  (3)  from  the  square  of  (1), 
—  T  -  /  s^y*—Axy  —  1 ; 

^  >-/^  X  ""-  /  x-{-y=xy  =  2±V5, 

-^J^^^l  x-y       =1; 

'/'z/r  whence,  ar=|(3=h»/5),  and  y=^(ld=f/5),  ^n*. 

49.  Let  X  and  y  represent  the  numbers ;  then 


x'-f=zy, 
x'-y'=x'^y\ 

(1) 
(2) 

Put  x=:vy]  then  the  equations  become 

vy-y*=vy\ 
vy^y'==vY-\-y\ 

(3) 
(4) 

Dividing  each  equation  by  y', 

(v«~l)y=vHl. 
(264) 

(5) 
(6) 

190 


QUADRATIC   EQUATIONS. 


From  (5)  we  have,  by  the  rule  for  quadratics, 

2v=ldbV^5. 
Multiplying  (5)  by  v,  v*—v=zv*\ 

transposing  in  (5),  v*=v-\-l  ; 

whence,  by  (8)  and  (9),       v*— v=v  +  l, 


or. 


l=r2y. 


(7) 

(8) 
(9) 

(10) 


Putting  this  value  of  (v*— 1)  in  (6),  we  have 

2vy=v'  +  l=i;  +  2, 
or,  4i'y  =  2v+4; 

from  (7),  2(litf^5)y==5±V^5; 

whence,  y=zt^V5 

and  x=v}/=^{l±:V5)  -  ^V5 ;  or         x=}(5±y5) 


VAns, 


ANOTHER    METHOD. 

«'—y'=^y» 

(1) 

^_y«=ar'  +  y'. 

(2) 

From  (1), 

^•-:ry=y'; 

(3) 

completing  square. 

.•-..?=¥• 

or, 

«-iy=±i»'s-y. 

or, 

x=i(l±Vo)y; 

by  involution. 

^'=i(3±»'5)y', 

Substituting  the  values  of  x*  and  x^  in  (2),  we  have 
(2±f/6y-y'=i(3±*/5y  +  y', 

(lztV5)y=^(5±y5)  ; 

x=^(5±y5) 


y-^^''       \Ans. 


(264) 


PROBLEMS.  191 


PROBLEMS    IN   PROPORTION. 

(334,    page  276.) 

6.  Let  X  and  y  be  the  numbers. 

ar  +  4  :  y  +  4=  3  :  4, 
ar  — 4  :  y— 4=   1  :  4. 
Therefore, 

Zy—4:Xz=z       4, 
y-4a;=-12, 
whence,  2y=     16 ;        y=8,    x=^b,  Ans, 

*1.  Let  X  and  y  be  the  numbers. 

a:4-y  =  27, 
xy  :  «'  +  y'=20  :  41. 

Multiply  the  antecedents  by  2,  and  we  have,  by  composition  and 
division, 

{x  +  yY  :    2ary     =81  :  40, 

(x-yY  :    2xy     =  1  :  40; 
whence,  (^+y)'  •  (^— y)'=81  :     1, 

or,  x+y     :    x—y   =  9:1, 

or,  27    :    x—y   =9:1, 

or,  x—y   =  3; 

x  +  y   =27; 
whence,  ir=15,     y=12,  Ant, 

8.  Let  a;=  the  number  of  gallons  of  rum, 

y=  "  "  "  brandy. 

x—y  :  y  =100  :  a;, 

a;— y  :  ar  =     4  :  y. 
By  Proposition  XI, 

{x-yY  :  iry=400  :  ary,       (co^^j 

ox,  X^-yY  •  1  =^0^  =  1 ; 

therefore,  x—y=  20. 

(276) 


192 

PROPORTION.      ff,  ^  ,^  ;  p   /  ;  /,^  /  jT y 

Again, 

1:^  =  25:-,        4^.,^,^C.^ 

or, 

1  :  25=    y*  :  x\                            y  ~i-S- 

1:5=    y    XX, 

therefore, 

6y=    X', 

■whence, 

y  =  5,     ar=25.  Am, 

9.  Let   a?+y=  the  greater, 
X — y=  the  less. 

{x  +  yY  =  x*^  3ar'y  +  3:ry«  +  y\ 
lx—yY=x^  —  dx'yi-3xy*—y\ 

Hence, 


6a:V  +  2y':  8y'=   61   :  1, 

3x'  +  y'  :  4y'=    61  :  1 ; 

therefore,                           3^'  +  y'=244y', 

a:'=    81y'. 

But,                                        «'-y»=320, 

80y'  =  320, 

y=    2, 

a:=   18; 

whence. 

20  and  16,  Ans, 

Note.— See  Example  4  for  a  diflferent  solution. 

__^ 

.^^ 

-x.^^    .      v~y 

10.  Let  X  and  y  be  the  numbers. 

^V  -    6  ^  ,    y-  -  JJ 

a-y  :    a:'  +  y»  =   2  ;  5, 

>^"7\'^r"  ^y-^.'i  ^-;  :! 

or,  22-y  :  a:'  +y'  =  4  ;  5  ;    ^V^-  ^Y '• -^v  V  ZyV  :  ^  /  / 

by  Prop.  VI,       (i^  +  y)':  (^-yr=  9  :  1.     ^f\^l\\);\r' 

We  have  given,  ar  +  y  =60  ;  ^^  '  ^  V  •'  •'  -^  'V 

whence,  by  (1),  ar-y  =20 ;  \^f^^f^       ^'  '    ( 

therefore,  a?=40,    y=20,  Ans?     7 

11.  Let  3ar  and  2a;  be  the  numbers. 

3x4-6  :  2ar--6  =  3  :  1, 
whence,  a;=8 ;  24  and  16,  Ans, 

12.  Take  16a;  and  9a;  for  the  numbers. 

16a;  :  24  =  24  :  9a;, 

x=  2;  therefore,     32  and  18,  Ans, 
(276-277) 


PROBLEMS.  193 

18.  Let  X  and  y  be  the  numbei's. 

x  +  y  :  x—y=     4:1; 
by  Prop.  VI,  x  :  y       =5:3; 

5i/=3x. 
Again,  a;'  +  y' :  x   =102  :  5, 

«'  +  5V^'  :  X  =102  :  5, 
or,  34^  :  25   =102  :  5; 

whence,  ar=15,   y=y,  -4w«. 

14.  Let  X  and  y  be  the  parts.  /  /  ^^  _  y  ///,'/ 

a:  :  y=   9  :  1,         v  -/-^  -  :^y 

9y=  a:;  /^^  //"  .    2^' V  ^  '^ 

a;  +  y  =  20;  i/ifyi^    =  ^ 

whence,  y=   2, 

a?  =18.  Hence,       V'2ry=6,  Ans, 

15.  Let  3j?  and  2ar  be  the  numbers. 

32?  + 6  :  2ar— 6=   9  :  4, 
:r=13; 
whence,  39  and  26,  Ans, 

16.  Take  ar,  ary  and  xy*  for  the  numbers. 

a;'y  :  2ry=ar  :  2a'y^ 
or,  1  :  y«    =1  :  2y; 

whence,  y=2. 

By  the  conditions,       ar  +  ary' = 300, 
whence,  5a;=300;  therefore,  60,  120  and  240,  ^n5. 

17.  Let  x  and  y  be  the  two  numbers, 

ar'  +  y'  :  a;»-y'  =  559  :  127; 


Prop.  VI, 

2ar'  : 

2y»=:686  :  432, 

or, 

X*  : 

y»  =  343   :  216, 

or, 

X  : 

y=      1  :     6; 

whence, 

6.r=   7.y. 

By  the  conditions. 

a:V  =  294; 
a;' =  7-49  ;  whence,  7  and  6,  Ann, 
(«77) 

194  PERMUTATIONS   AND   COMBINATIONS. 

18.  Let  X  and  y  be  the  numbers. 

x'  :  f=  S  :  1,  (1) 

,         ,  y'  :  ar'=96  :  1;  (2) 

from  (3),    7  ^  ^^  y-^, 

y  :  ^  : :  3  :  /  i     a 

_  ^       whence  from  (4),  x'=S^'9Qx% 

-^Vor,  ir^= 3^-96, 

or,  ar»=3'-(32)' ; 

therefore,  a:=12,  and  y=24,  ^n#. 

19.  By  Prop.  X,  this  proportion  becomes 

(^  +  1)'  :  (•^-1)'=2  :  l; 
whence,  xi^2—^2=x-\-l, 

and,  ^=f^2^?  ^'*'' 

20.  Putting  the  given   equation   in  the  form  of  a  proportion,  we 
have 

a-\-b  +  c-{-d  :  a—b  +  c—d—a-^b—c—d  :  a  —  b—c-\-d\ 
by  Prop.  VI,  and  dividing  by  2, 

a-\-c  :  b-\-d=a—c  :  b—d, 
or,  a-\-c  :  a  —  c=2b  +  d'.  b—d^ 

Again  by  Prop.  VI,  a  :         c=b        :  rf, 

or,  a  :        b=zc         :  c?,  Ans, 


PERMUTATIONS  AND  COMBINATIONS. 
(341,  page  283.) 

I.Here        n  —  r  +  l  =  10  — 4  +  1  =  7  ;  hence,  ^t>r7 

10- 9- 8- 7=5040,  ^n*.  '     ^ 

2.  6-5-4-3-2-l=V20,  u4n«. 

(277-283) 


PERMUTATIONS   AND   COMBINATIONS.  195 

3.  By  formula  (B), 

10'9*8-7-6-5-4*3-2*  1  =  3628800,  Ans, 

4.  Omitting  the  0,  the  other  four  figures  can  be  arranged  in 

4  •  3  •  2  •  1=24  ways. 

Now  we  must  reject  every  combination  formed  by  placing  the  cipher 
he/ore  all  the  other  figures.  Hence,  in  each  of  the  24  combinations 
of  the  figures  4,  3,  2,  1,  the  cipher  may  have  4  places.     Therefore, 

24x4=96,  ^n*. 

5.  By  formula  ((7), 


6.  Formula  (C)  applies ;  n  =  16,     r  =  5j  n— r+l  =  12. 

„     16-15-14-13-12      _^„      . 
^=       5-4    3-ii-l-=-*^'^'-^"'- 


^    „  .»     20 -19 -18 -17 -16 -15      „_^„     . 

V.Here  2^_______=38760,  ^«, 


8.  Omitting  the  boy  denied  the  privilege  of  the  head,  the  others 
can  be  arranged  in 

5  •  4*  3  -2  •  1  =  120  ways. 

The  omitted  boy  may  occupy  each  of  the  five  lower  positions ;  hence 
all  the  ways  will  be 

120x5  =  600,  Ans, 


9.  The  prime  numbers  below  40  are  1,  2,  3,  5,  7,  11,  13,  17,  19, 
23,  29,  31,  37  ;  or  there  are  13.     Hence  by  (  341), 

n=13,     r=!i^=6,     n-r+l=8. 

„     13  •12-11  •10-9'8     __-     . 

Z= ; — =1710,  Ans, 

6- 5-4-3-2-1 

(284) 


196  PERMUTATIONS   AND   COMBINATIONS. 

10.  By  formulas  {A)  and  (C^,  we  have 

«(n-l)(»-2)(»-3)(n-4)=.120(fc^'^)  ;    ' 

or,  11*  — 7  n—     8, 

whence,  n=      8,  Ans. 

11.  Heren  =  8;  and  by  (  341  ),  r=:- =4,     n—r  + 1  =  5. 

Hence,  «  =  -————  —7^  ;  and  -—=$0.60,  Am, 

4  *  o  *  ^  *  1  70 

12.  By   formula  (C),  if  we  let  n—  the  number  of  horses, 

n(n-l)(n-2)     ^/«(»-l)\ 

3-2-1  H     2-1    r  ''^'  ''"'^'        *• 

13.  First  omit  4  of  the  points  which  are  in  the  same  straight 
line ;  and  considering  the  remaining  8,  we  find  the  number  of  com- 
binations of  8  things  taken  2  at  a  time  to  be 

8-7 

2-n  =  28; 

or  28  different  straight  lines.  Now  the  four  other  points  may  be 
joined  each  with  the  7  points  not  in  the  same  straight  line,  making 
28  more  different  straight  lines  ;  and  adding  the  line  containing  the 
five  points,  we  have 

28  +  28  +  1  =  57,  Am. 
Again,  if  no  three  of  the  points   are  in  the  same  straight   line, 
there  will  be  as  many  straight  lines  as  there  are  couibinations  of  12 

things,  taken  2   at  a  time,  or  =66.     But  since  five  of  the 

5  •  4 
points  are  in  the  same  straight  line,  ——  =  10  of  the  combinations 

are  lost ;  and  adding  the  straight  line  containing  the  five  points,  we 
have  66—10  +  1=57,  Am. 

Generally,  if  there  are  n  points  in  a  plane,  of  which  J9  are  in  the 
same  straight  line,  there  will  be 

n{n-\)     pi^p-X) 
2-1  2-1 

different  straight  lines  formed  by  joining  the  points. 

(284) 


ARITHMETICAL   PROGRESSION.  197 


ARITHMETICAL  PROGRESSION. 

(353,   page  288.) 

Note. — In  some  of  the  following  examples,  we  shall  employ  the  formulas  of 
354,  instead  of  substituting  the  given  data  in  the  primary  equations. 

1.  /=7  +  35 -3  =  112,  ^ns. 

2.  «=Y '280  =  6440,  ^n«. 

3.  Here  we  have  «,  n  and  d  given  to  find  a  and  I.    From  formulas 
(A)  and  (5), 

l—a=.{n  —  \)d, 

2* 
l-\-a= — .  Hence, 

n 

_2s—n{n—l)d  2s-\-n(n—l)d 

and  by  substitution,  a=:2,  /=37,  Am, 

4.  Substitute  the  value  of  /  from  formula  (A\  in  formula  {B\  and 
we  have 

.=l\2a  +  (n-l)dl  =151(2 +  ~)=2626,  An>. 


37  —  7 
6.  By  formula  ((7),  (352),  d~ — - — =6,  and  the  terras  are 

5 

7,  13,  19,  25,  31,  37,  Arts, 


6,  We  have  a,  n  and  8  given,  to  find  d  and  Z.   By  No.  5,  (354), 

7     ^440     ^     ,^,       ,     (3720- 180) -2     ^     . 

7.  Here  »=  11;  hence, 

5=V(9  +  109)=649,  ^»«. 
(288-289) 


198  SERIES. 

8.  By  (352),  rf=i^=5V,  ^»«. 


9.  By  No.  1,  (354), 

a=i|i{2+(365-l)2i=365'=$1332.25,  ^n*. 

10.  By  9,  (354),  since  we  have  given  a,  d  and  s  to  find  n, 

3_40±V'(37V  +  8-3-438      -37^:^11881      — 37±109      ,„ 

„= LJ _= = 6— =''• 

Ans, 

11.  The  last  term  will  be  n  ;  and  by  formula  (B\ 

«=^(l+n),  ^n«. 


12.  Herea=l,  rf=2,  and  *=^{2 +  (n  — 1)2J  =n',  Ans, 

2 


13.  We  have  n,  rf,  and  «  given ;  hence, 

1900  — 24-25-3      1900  —  1800 


60  60 


2yAn8, 


14.  We  have  given  a,  rf,  and  «  to  find  n.     By  No.  9,  (354), 


«-  i  -        ^        ; 

3  3 

—  1  4-  A4J. 

or,  w= — 5 3_  =  i50,  Ans, 

T  a.  ' 


,^    666-66     ^      ,  ,6666-666     ,^^^ 

15.  =5=rf;  and =1200. 

1  ^U  o 

Hence,  w=1200  + 120 +  14  =  1334,  Ans, 

(289) 


ARITHMETICAL   PROGRESSION.  199 

PROBLEMS  IN  ARITHMETICAL  PROGRESSION. 
(355,  page  291.) 

1.  Let  (^— y),  ^  and  {x-{-y)  be  the  numbers. 

From  first  condition,  3x  =   18, 

X  =  6  ; 
from  second  condition,  3x'  +  2y'  =  1 5  8  ; 
whence,  2y'=:  50, 

y  =     5.  1,  6  and  11,  Ans, 

2.  Let(a:— 2y),  (x—y)y  «,  («  +  y),  (ar -f  2y)  be  the  numbers. 

From  Ist  condition,  5x  =     65, 

a;  =     13 ; 

"     2d         "  5a;'  +  10y'=:1005, 

a:'  +  2y'=   201; 

whence,  2y'=     32, 

y  =       4.        6,  9,  13,  17,  21,  Ans. 

3.  Let  (a;— 6),  (a;— 2),  (a; +  2)  and  (a? +  6)  be  the  numbers. 

(a:'-4)(a;«-36)=a;*-40a:'  +  144  =  l'76985, 
or,  a;*-40a;'= 176841, 

a;«_40a;'  +  400  =  l77241, 
a;»  =  20i:421, 
a;'=441, 
a;  =21 

15,  19,  23,  27,  Ans, 

4.  Let  (a;— 3y),  (ar— y),  (X'{-y)  and  (a;  +  3y)  be  the  numbers. 

2aj=   8, 

or,  ar=  4; 

aj'-y»  =  15; 

whence,  y'=  1, 

y  =  1.  1,  3,  5,  7,  Ans, 

(291-292) 


200  SERIES. 

6.  Let  n=  the  number  of  days  the  first  person  travels, 
d=lj         and  l=:l  ■^{n  —  l)d=n. 

.    «=q(1  +  n)  =  the  whole  distance. 


and 

15(w-6)=           «             " 

hence, 

in{l+n)  =  15(n-6\ 

or, 

n'  +  w=30n— 180, 

n'-29n=-180. 

First  person  travels. 

n  =  Y=t-V-=     9  or    20, 

-6       -6, 

Second     «         ** 

3  or    14,  Ans, 

Explanation. — Call  the  first  person  A,  and  the  second  B.  Now  B  overtakes 
and  passes  A  after  A  has  traveled  9  days  and  B  3  days.  But  as  A  is  increasing 
his  rate  one  mile  per  day,  he  finally  gains  on  B,  and  overtakes  and  passes  him 
after  A  has  traveled  20  days,  and  B  14.  They  are  together  after  having  trav- 
eled 45  and  201  miles. 


6.  Let  x=:  one  of  the  equal  payments. 

At  the  given  rate  per  cent  $60  will  amount  to  $61  at  the  end  of 
60  days.  As  the  rate  of  interest  is  ^-^^^  of  the  principal  for  a  day, 
the  firet  partial  payment  will  amount  to 

the  second  to  ^+3loo-^» 

the  third  to  ^+3lJo^, 

and  the  last  to  x. 

Hence  the  sum  of  the  partial  payments  is, 

Q0x-\-^j\^{59x-\-58x  +  5lz. . . .  +x), 
or  by  summing  the  series, 

60^  +  tVo^' 
Since  the  debt  is  to  be  canceled,  we  must  have 
60a-  +  T¥o^=     61, 
T259ar=7320, 

whence  ^=^1t2J7»  ^^' 

(292) 


ARITHMETICAL   PROGRESSION.  201 

7.  Let  {x—Sj/y  (j^— y),  (^  +  y)»  and  (x  +  Sy)  be  the  numbers;  then 


2j:'  +  18y'=66, 

(1) 

2«'  +  2y'=61, 

(2) 

y'=h 

y=i, 

*=V; 

4,  5,  6, 1,  Am. 

:  notation  as  before, 

4a;=24, 

(1) 

■10j:y  +  9y*  =  945. 

(2) 

Whence,  ar=6; 

and  from  (2),  y*— 40/=  -39, 

y'=      20±19, 

y=ly  3,  5,  7,  9,  -iw*; 

9.  Let  (ar—y),  a^,  and  (^  +  y)  be  the  digits.     We  have 

I00(x-y)  +  10x  +  x  +  y^^^^ 
ox 
100{x—i/)  +  l0x-hx  +  i/-{-l98  =  l00{x  +  y)  +  lt)x  +  x—y\       (2) 
from  the  second  equation,  y=l» 

from  the  first,  3jr=9y, 

x=3 ;  234,  Ans. 

10.  Let  n=  the  number  of  days.     Then  from  No.  2,  (3«54:), 

-[6  +  (n  — 1)2}=  the  distance  A  travels, 
2 

^{8  +  (n-l)2}=  «  B      " 

The  sum  of  these  expressions  is  equal  to  the  whole  distance ;  hence, 
2»'+5»=102, 

2n=  — fit.3/;  n=6,  Ans. 

11.  Let  a;~  the  number  of  weeks  if  no  one  dies; 

21a?=  the  pecks  of  corn  distributed.  ,  ' 

In  the  second  case  the  number  of  pecks  distributed  each  week 
will  form  an  arithmetical  series,  in  which 
.  (292) 


202  SERIES. 

0=21,      cf=--l,      n=2aj; 
and 

S=zx(i2—'2x-\-l)=z  the  pecks  of  corn. 

Equating  the  two  expressions  for  the  number  of  pecks,  we  obtain 
43jr-2ar'  =  2l3-, 
2a:=22, 
z=ll  ;  21a;=231,  Ans, 


GEOMETRICAL  PROGRESSION. 


(364,  page  296.) 


1.  Here  a=l,  r=2. 


^     2»-l     512-1      ^^^      . 
^=-— ^  =  — =511,  Ans. 


2.  a =2,  r=3 ;  hence, 
/=:2-3'  =  2-2187  =  43Y4,  ^n«. 

3.  a=l,  r=§ ;  hence, 
^="131-=      ^-1-69049  ^  -59049  '  ^'^• 

4.  r=(VJ)^=2.     Hence,  48,  96,  ^w«. 
6.                       r=(ifi)^  =  (256)^  =  2.     Hence, 


6,  12,  24,  48,  96,  192,  Ans.  o  ^^ 


6.  a=l,  ^=f;  hence, 

*I,  a=f,  »*=f ;  hence, 

^      s 

(292-297) 


GEOMETRICAL   PROGRESSION.  203 

8.  a=5,     r=i ;  hence, 

1         3 


JL?. 
5-       •"" 


10.  «=f'o'o>     rrrr-j-i^  ;  hence, 

11.  «=i»     '*=— i;  hence, 

12.  o=},     r=-i;  hence, 


X 

13.  o=l,     r=- ;  hence, 

a 

^        I  la, 

,  Ans, 


X     a—x     a — X 
a        a 


1  X* 

14,                     o=-,  r= — i;  hence, 

a  a 

1 

a  1       a'     __     a 


^4 


15.  By  formula  {B\ 

1785=^^^  ~   ^  =  255a.     Hence,     a='7,  ^w.<r. 
(297) 


204  SERIES. 

16.  By  formula  (5'), 

7812^^'  +  " 


or,  /= 


5-1' 
31248  +  a 


By  (B),  Y812="(f    /^=3906a. 

5  —  1 

Hence,  o=2,  and  /=6250,  Ans, 

17.  By  formula  (^), 

1215  =  5r', 
or,  r'=243.     Hence,  r=3,  Ans, 

18.  By  formula  (5), 

2"-l     1024-1     ,^„„     . 
/Sf=-— — -= =  1023,  Ans, 


PROBLEMS  IN  GEOMETRICAL  PROGRESSION. 
( 365,  page  300.) 

3.  Let  the  numbers  be  ar,  Vxy,  y  ;  then 

X  -\-V^  +  y=   21,  (1) 

x'+   a:y  +  /=:189,  (2) 

The  solution  is  like  that  of  Example  1.    We  have  a=21,  5=189. 

441-189     ^ 
i^=— ^^— =6; 

and  from  (1)  and  (2),     ar4-y=15, 

x—y=z  9.     Hence,  3,  6,  12, -4nf. 

4.  Let  ar,  ary,  ary'  be  the  numbers ;  then 

a;4-ary  +  a:y'  =  210,  (1) 

xy^-x=   90.  (2) 

(297-300) 


GEOMETRICAL   PROGRESSION.  205 

Dividing  by  ar,  and  eliminating  y',  we  have 

120 

*=:^ • 

24-y 

Substituting  this  value  of  x  in  (2), 

12y'-9y=30, 
y=  2, 
ar=:30  ;  30,  60,  120,  Ans. 

5.  Let  Xj  ary,  ary',  a?y"  be  the  numbers ;  then 


x^xy+xf  +  xt/'=80, 

(1) 

xf     _4 

xy-\-X]/^     3  * 

(2) 

From  (2), 

f   _4 

1+y    3' 

or. 

y=2; 

whence  by  (1), 

aj+2a:  +  4a;  +  8j;=30. 

a:  =  2; 

2, 

4, 

8, 

16,  Ans, 

6.  Adopt  the  same  notation  as  before  ;  then, 

ar  +  a-y'  =  148,  (1) 

ary  +  ar/  =  888.  (2) 

Dividing  (2)  by  (1),  y=     6, 

from  (1),  37a;=148, 

x=     4 ;        4,  24, 144,  864,  Arts. 

7.  Solution  the  same  as  in  1  and  3. 

8.  Let  X,  ary,  a?y',  xy*  be  the  numbers ;  then, 

xy^-xy=  24;  (1) 

ar+ary*  :  ary-hary'=     7  :  3, 
or,  1  +  y'  :    y+  y"=     7  :  3, 

dividing  by  y  +  1,  y'— y  +  l  :     y=     7:3, 

3y'-10y=:-3;  (2) 

from  (2),  y=     3, 

whence  from  (1),  a;=     1 ;        1,  3,  9,  27,  Ans. 

(300) 


206  SERIES. 


9.  Let  ar,  jy,  xy*  be  the  numbers ;  then, 

« 

a:+^y  =  20, 

(1) 

xy*—xy=z^O. 

(2) 

,.  ..                                 1+y      2 
r  division,                                 ^,_^-  -, 

^y*-5y=  3; 

y=  3, 

;r=   5; 

5,15, 

45,  Arts. 

10.  Adopt  the  same  notation  as  before;  then, 


. 

a:y  =  216, 

0) 

x^ 

+  ary  =  328. 

(2) 

From  (1), 

xy=     6, 
x^-   '^• 

• 

from  (2), 

,      328 

hence, 

»/ 

-82y'=-9; 
y=    3, 

ar=     2; 

2,  6,  18,  Ans, 

11.  Let  XyVxy^  y  be  the  numbers  ;  then 

.    '  a;+V^4-y=13,  (1) 

(:r  +  y)*^=30.       _  (2) 

Hence,  a;4-y=13— Vary, 

30 
Vxy 
ary— 13  V^=:— 30, 
\^=     3, 
xy=     9, 
x-\-y=     10, 

x^y=     8  ;  1,  3,  9,  Ans, 

(300) 


GEOMETRICAL   PROGRESSION.  207 

12.  Let  Xy  xy^  a;y',  be  the  numbers;  then 


xY=  64, 

(1) 

«'  +  a;yf  a:y  =  584. 

(2) 

From  (1), 

by  substitution  in  (2), 

16/-130/  =  -16, 
y=     2. 

whence, 

x=     2; 

2,  4,  8,  Ans, 

13.     Let  ar,  rry,  a:y'  be  the  numbers  ;  then 


xy=i; 

and, 

xy—x  :  xy^—xy=5  :  1, 

Of, 

y-1  :    y'-y=5  :  1, 

or. 

1  :             y=5  :  1, 

whence, 

y=}; 

from  (1), 

ary=l, 

(1) 

(2) 
ar=5  ;  6,  1,  ^,  ^/w. 


14.  We  might  divide  120  into  four  parts,  .which  should  be  in 
arithmetical  progression,  by  assuming  a  sinyle  unknown  quantity  for 
the  first  term,  or  the  common  diflference  ;  but  as  we  have  two  condi- 
tions to  satisfy,  we  take  two  unknown  quantities.  Let  x  be  the  first 
term,  and  y  the  common  diflference  ;  then 

x  +  {x  +  y)-\-{x+2y)  +  {x-h3y)  =  l20, 
or,  2a:  +  3y=60.  (1) 

Again,  we  have  by  the  second  condition, 

X,  {x  +  y-12),  (ar  +  2y-12),  (ar4-3y  +  24), 

for  the  geometrical  series.     Hence,  by  (  363 ), 

{x  +  y-l2y=x(x-^2y-l2),  (2) 

or,  y'— 12a:~24y=  — 144; 

substituting  from  (1 ),  y*  —  Qy—     216, 

-?y,ic- V.  v^;r.  y^-?/  y=     18, 

Ult/2'o,^z3ef  ^^     3.         3, 21, 39, 67,  ^w*. 


-/ 


208  SERIES. 


15,  Let  ar,  Va;y,  y,  be  the  numbers  ;  then 

ar+^+y=31,  (1) 

a:  +  y  =  26,  (2) 

V^=  6, 
xy=25, 
whence,  jr— y  =  24, 

ar  +  y  =  26;         1,  5,  25, -4yw. 

16.  Let  Xj  xy^  a-y',  a-y*,  ary*,  ry*,  be  the  numbers  ;  then 

ar  +  ary  +  ry'4-ary'+a;y*+ary*  =  189,  (1) 

xyJfxy'=   54,  (2) 

from  (1),  a;(l4-y  +  y')+^y'(l+y  +  y')  =  189, 

189 
x-\-xy*  = r, 

^     1+y+y" 
Multiplying  this  equation  by  y,  and  equating  with  (2), 

189y 


54: 


1+y+y" 


2y«-5y=-2, 
y=    2, 
ar=     3; 
3,  6,  12,  24,  48,  96,  Arts, 

17.  Adopting  the  same  notation  as  before,  we  have 

(a;+a5y)  +  (ar+ary)y*=189-36  =  153,         (1) 
(a; +  a:y)y«  =  36.  (2) 

1+y* 
By  division,  — ^i— =  y , 

whence,  y=2, 

a;=3 ;  3,  6,  12,  24,  48,  96,  Ans. 

18.  Let  X  represent  one  of  the  equal  payments ;  then 

p{\  +r)—x^  due  after  1st  payment^ 

^(14-r)'-ar(l+r)-a;,  «        2d 

^(I4.r)'-a:(l+r)'-ar(l+r)-a;,  "        3d         " 

etc.,  etc.,  etc., 

(801) 


DECOMPOSITION    OF    FRACTIONS.  209 

or,  ;>(l  ■\-ry-x{l  -hr)'^'- -x{l-\-r)-x, 

due  after  the  nth  payment.     And  since  the  debt  is  to  be  canceled  by 
the  nthy  or  last  payment,  we  have 

p{l-^ry—z(l+ry-'—x(l  +r)'-'— — a-(l  +r)—x=0  ; 

or,       ;>(l+r)"-{(l+r)'->+  (1 +r)'-'+ . . . . +(1 +r)  +  l(ar=0. 

Summing  the  series  in  the  parenthesis,  ^    /  /-/ r~/  y- 

whence,  X=^rf7^f »  ^"*- 


DECOMPOSITION   OF   RATIONAL  FRACTIONS. 
(369,   page  308.) 


1. 

x'-^x-V\^  =  {x-n){x-^\ 

Hence  we  assume. 

7^_24      _    A    ^    B 

--'.  -s?t--i^.r»'''~' 


(ar-7)(2:-2)      x-1     x-2' 
whence,  1x—24  =  {A-h  B)x—(2A  +  IB) 

Equating  coeflScients, 

A+£=  1,      -i</^  J^  .  _ 

whence,  <i^A  A-  5,     £=i.  ^  ^  /  'V 

2.  2x^  +  Sx--20  =  (2x—5)(x-\-4),  -^  ^.^J^ 

.  20X  +  2  ^      ,     ^  7 

^^'"^  (2:r-5)(^  +  4)-27-r5-^JT4- 

(301-308) 


210  SERIES. 

then  .     20x+2  =  (A-h2B)x  +  (iA-BB), 

Equating  coefRcienis, 

^  +  2^=20, 
4A-5B=  2; 
whence,  ^=8,     i?=6. 


+ 7, -4n«. 


2a;— 5     x  +  i 


3.  Assume 


6ar«-22  +  18      _    A        ^_        C 


(x— l)(x'  — 5j:  +  6)     x—\      x  —  2     x— 3' 
then 
ea:*— 22a:4-18=^(x-2)(x-3)+^(x-l)(x-3)+C(x-.l)(«-2). 

Hence  we  have 

A-[-B+C=  6, 

5^ +  4^4- 3  C=  22, 

6^  +  3j5+2C=18. 
Whence,  .4=1,     J?=2,     C=3. 

H ^  + r,  Ans. 


x—\     x—2     x— 3 


4.  Let  _:_=:_+__-  + 


ar'— a?     a;      a:  +  l     ar— l' 
then  ar  +  2=^(x'-l)  +  ^(a:»-ar)4-C(a;«  +  a;). 

Thence  we  have 

A-\-B->rC=     0, 
-^+C=     1, 
-^=     2. 
Whence,  ^=—2,     j5=:^,     (7=1. 

2^        1       ^       3 
"x'^2(^Tr)"^2(^=T)»  "*''*• 

(308) 


BINOMIAL   THEOREM.  211 

6.  We  have 

Hence  we  assume 

10  A  B  C         D 

+ ^  +  — ^  + 


a:*--13x'  +  36      x^2     .r  — 2     a:  +  3      a:— 3  ' 
from  which  we  find 

10=     ^(a;'  — 2.c'  — 9^+18)  +  ^(x»+2ar'  — 9a;— 18) 
4-C(x'-3a:'-4a;+12)+i>(a;'H-3x«-4ar-12). 

And  equatinpj  coefficients, 

A  +  B+C-{-D=  0, 

— 2^  +  2^-3(7+3i)=   0, 

9^  +  9^  +  4(7+4i>=:  0, 

18^-18^  + 12  C—12i>=:10. 

Hence,  combining  the  first  and  third  equations, 
A=—B,     and  C=—D, 

From  the  second  and  fourth  equations,       u£i^'iztvci^ 

2J-32>=0,  . 

18^  — 122>=5. 

Hence,  A=\,     B=-\,     (7=-i,     D=\, 

1111 

+  —f —.  ,  An9» 


2(«  +  2)     2(.r--2)     3(i;+-3)     3(ic— 3) 

BINOMIAL  THEOREM. 
(377,  page  316.) 

In  examples  1  to  10  the  exponents  are  whole  numbers,  so  that 
these  examples  need  no  solution  here.  In  example  10  the  numerical 
coefficients  are  the  same  as  those  of  example  2,  of  the  illustrations ; 
and  we  shall  have  the  answer  by  simply  changing  +«  to  —x  and 
observing  that  the  odd  powers  oi  —x  are  negative. 

(308-316) 


212  SEBIES. 

11.  Here  we  have 

A  =  +1  ■ 

B  =  Axn  =  +1 

2  3    3    2  3  '6 

_         „     n— 2               2       5    1  2-5 

i>=Cx— -— =H •-•-       =  + 


i:  =  j) 


3  3-633  3-6-9 

»-3  2-5       8    1  2-5-8 


4  3-6 -9    3    4  3-6-9-12 

The  law  of  the  numerical  coefficients  is  now  evident,  and  we  may 
write  out  the  series,  observing  that  the  odd  powers  of  — x  are  nega- 
tive.    Hence, 

,,        A     ,      X      2x«      2-6j:«       2-5-8x«         2-5-8-lla:' 
(l~arf  =  l--  - 


3     3-6    3-6-9      3-6-9-12     3-6-9-12-15  * 

Ans, 

12.        A  =  +1 

B  =  Axn  =  +1 

n-1  13    1  3 


C  =  Bx 


2  4    4    2  4-8 


J,       ^    n-2         ,3      7    1  3-7 


E=Dx 


3  4-8    4    3  4-8-12 

n-3  3-7       11     1  3-7-11 


4-8-12     4     4  4-8-12-16 


Hence, 


^         ^  4        4-8       4-8-12       4-8-12-16 

_  T/i  ,  i 2  3-7 3-7-11 . 

""**  ^    "^40     4^8^     4-8-12a'     4-8-12- 16a*"^  * '  *  *''' 

Ans, 

13.  Since  ?i  is  the  same   as  in  Example   11,  the  numerical  coeffi- 
cients will  be  the  same.     Hence, 

/       rA       4     a~^^     2-a~36'     2  •  5a~'6'     2-5-8«~'3'6* 
^    ^   ^  ^3  3-6  3-6-9  3-6-9-12 

\    ^^a     3-6a'^3-6-9a'      3  •  6  •  9  •  12a*^  /' 

^  (316) 


BINOMIAL   THEOREM.  213 

A  =  +1 

B  =  A  X  n  =  ^1 

C  =  Bx'^=(-l)x(-l)  =  +l 
D=Cx''^  =  (  +  l)x(-l)  =  ^l 

U=J)x  !^  =  (_i)x(-l)=  +1 

Hence,  (a—b)-'=a-'  +  a-'b  +  a~'b'-\-a-'b'  + 

I      b      b'     b' 

=  -+-7+-,  +  —+....  ,  Ans, 
a     a'     a'     a* 


15.  — — =cr(l— a;)"'.     Omitting  the  factor  a, 

A  =+l 

B  =A  xn        =  (  +  l)x{  — 2)  =-2 

C  =  B  X  _-=(-.2)x(-|)=.+3 

j^  = />  X  ^^^  =  (-4)x(-f)  =+5.     Hence, 
a(l  — a;)-*^a(l  +  2a;  +  3x'  +  4.c'  +  5x*  +  6u;'  + ),  ^n«. 

16.  The  coefficients  are  the  same  as  in  Example  10. 

17.  A  =  +  1 
B  =  Axn        =(4.i).(4-f)                    =4-| 

(316) 


214  SERIES. 

^       _,     n-S       I       2-4    \     /     1\     /1\  2.4.7. 

Hence, 

,xl       I     2a~V     2a"3c*     2  •  4a~V     2  *  4  •  7a~^3^c' 
(a  — c')'=a' 


3-6         3-6-9  3-6-9-12 

"""  \        3a"~3-6a'""3-6-9a'     3*  6*  9 '120*  /'      ***' 

18.    ^  =  =+1 

B=Axn        =(  +  l).(-i)  :=z-^ 

3 
=  +2^4       ■ 


5_ 


--¥=(-y(-=)(y 
-=-"-^-(-i^y  (-D  o-^i^. 

Hence,  we  have 
./  ,        ,^-i       y    ,     c-'x'     3c-V     3  •  5c-' j:'     3  •  5  •  7c-».r'  \ 

rf/        a:'         3j:*  3  •  5x'  3  •  5  •  7.c'  \ 

=cV"2?  +  ^^*-2"^T^c"^  +  2-4-6-8c«~- '  "h  "^''** 

_i 
Note. — The  exponent" of  c  in  the  first  term  is  (c')    ^  =:c  ' ;  and  the  ex- 
ponents of  c  in  the  terms  following  diminish  by  2  throughout. 

19.  ^  =  +1 

B  =  Axn  =  -3 

i)=Cx^={  +  6)x(-5)    --10 

^  =  i>x^=(-10)x(-y=:  +15 

Hence  the  law  of  tlic  coefficients  is  evident,  and  we  have 

(1  — a)-*=  1  +  3a  +  6a'  -f  10a'  +  15a*  +  21a'  +  28a"'  +....',  Ans. 
(316-317) 


'  DESITYl 


BINOMIAL   Td^EOREM.  ^       ^^  V*^''215? 


20.    A  =  +1 

B==Axn         =(  +  l).3  =4-1 

_         3 


5 

12 


9 
T6 

Hence, 

— '  —1  _i  _  i_i 

,x?^       4     3a  ^a;'     3a   ^x*     S'5a  *x'     3*5-9a    ^  x' 

(a'— ar  )*=a* .  . . . 

^  ^  4  4-8  4.8-12        4-8-12-16 

_      /       3a:'        3a:*  3* 5a;*  3*5  •Oa:"  \ 

~      \  ~3a'~*4  •  8a'~"4  •  8  •  12a'~4  •  3  •  12  •  16a' ~  /' 

Ans, 

3  3 

Note. — ^The  exponent  of  a  in  the  first  term  is  (a')  *  =.  a  - ;  the  exponents  in 
the  terms  following  diminish  by  2  throughout. 


21.  A  =  +1 

B=zAxn        =(-{-i)x(-4)     =-4 


C  =  Bx'^=(-4).{-l)  =+10 
i)=(7x!^=(  +  10)x(-?)  =  -20 
JS  =  Dx^^=  (-20)x(-^)=  +35 


F=zl!x~  =  (  +  35)x(— ^)  =  -56 

Hence, 

(a  +  y)-*=a-*-4a^V  +  10a-y-20a-y  +  35a-y-56a-V  +  .. 

1      4y     lOy-     20y'     36/     66/  . 

=a*-"T*  +  -^— ^^-^^ ^+....,^«*. 

(817) 


216  SERIES. 


r  -_i 

22.  We  have  =  r(l— r)^; 

V  1— r 


A  =+1 

"  =  -•^  =  (4)  •(-!)■©       -.^. 

4  \     2-3-5V     V       5/      \4/         ^2-3-4-5* 

Hence, 

/,        ^-i       /,      »•       6r'       6-llr*      C'll'lCr* 
r(l-r)   ^=.(l+^  +  2-T^  +  2-^.+  2^3T4-7^+.... 


=»•  +  -  + 


6r»       6-llr*      C-1116r' 


5      2 


^•  +  2^^T'-^^^3^.^+---^'^*' 


23.  Vi_a:*  =  (i_a-y\ 

^  =  +1 

n  — 2 


i>=  Cx 


~  \     2-15V  *  Vis)  *  W  ~ 


2 -IS" 

14-29 

2-315* 

14-29-44 

3 
rr        r.     ^-^        f.    14-29  \     /     44\      /1\  14-29-44 

Hence, 

»/ a;*      14j:"       14-29x"      14-29-44x'*    . 

VI «♦  —  1 >4«j? 

~        15     2-15'     2-3-15'        2-3-4-15^       ....,^'**. 

In  all  expansions  of  this  kind,  the  chief  thing  to  be  aimed  at,  is  a 
simple  and  systematic  method  of  calculating  the  numerical  coeffi- 
cients, and  one  which  will  clearly  show  the  law  of  their  formation. 

(8X7) 


BINOMIAL   THEOREM.  217 

(378,   page  318.) 

1.  (a~26)*=a»-3a'(26)  +  Sa(2by  +  (2&)* 

=a'  — 6a'6-fl2a6'  +  86',  Am, 

2.  (2a  4-  ^xy  =  (2a)*  +  4(2a)'(3a?)  +  6(2a)'(3x)'  +  4(2a)(3x)'  +  (3a:)* 

=  16a*  +  96a'x  +  216aV  +  216ai;'  +  Six*,  ^na. 

,.  (-iy=.-.©«6)'-,©V(jy 

=  1— 2a  +  |a'— la'  +  j^^«*,  ^ns. 

4.      (a'— tfj:  +  a:')*  =  («'— aa-)*  4-4(a'  -ax)  V  +  6(a'— aj;)'a;*  + 

4{a*—ax)x*-\-x\ 
Performing  the  operations  indicated, 
a'— 4a'a;+    6a  V—   4a  V  4-      a*x* 

4-   4aV— 12aV  +  12a*a:*—   4a V 

4-  ea*x*  — 12aV4-    6a V 

4-    4aV  — 4ax'4-a;'' 

a*— 4a'a:4-  lOaV  — 16ttV  4-  19a*x*  — 16aV4-  lOaV  — 4aa;'  +x% 

Ans. 

6.  The  numerical  coefficients  will  be  the  same  as  those  in  Exan> 
pie  12,  (377).     Hence, 

(4a'-3x)^ 


-(ia^\^      (^^')V)      3(^^')   M3^r      3-7(4a/')   '^(3x)» 
^      '  4  4-8  4-8-12 


')  ^(3a;)     3(4a')  ^(3x)'     3  •  7<-^-^^ 

~~4  r^8 

f       3x              3(3x)'  3''7(3i:)» 

^^a^ 3 1  j\ 

A{2ay      4 -8(20)^     4-8-12(2a)  = 


(318) 


218  SEBIES. 


FRENCH'S  THEOREM. 


(379,  page  320.) 

b      5 
1.  Here  we  have  n= 4,     a=2,     6  =  5,    -=-;  hence, 


c,  = 

(2)* 

=   16 

C,   = 

16" 

\ 

•i 

==  160 

0,= 

160- 

f 

•f 

=     600 

C,= 

600 

•?- 

•t 

=  1000 

^,  = 

1000' 

i 

•f 

=  625 

Hence, 

(2ar+ 

5yy-- 

=  16j:*  +  160j:V  +  600x' 

y'  +  1000jr/  +  625y*, 

Am, 

a.  n=5,     a=:2,     6  =  3,     -=-. 

'     a     2 

C,   =        (2)*  =       32 

C,   =       32  •  i  •  f  =     240 

(7,  =     240  •  A  .  1  =     720 

C^  =     V20  •  f  •  f  =  1080 

C,  =  1080  •  i  •  I  =     810 

C,  =     810  •  i  •  a  =     243 
Hence, 

(2a  — 3x)'  =  32a*~240a*j:  +  720aV-1080aV  +  810aa:*  — 243ar*, 

Ani^ 


s. 

n=6,  a=3,  6=4,  ^-=\. 

C,  =   (3)'      =   729 

C,  =   729  •  f  A  =  5832 

^3  =  5832  •  f -i  =  19440 

C^  =  19440  •  f  •  i  --=  34560 

C,  =  34560  •  ^  •  i  =  34560 

C.  =  34560  •  f  •  i  =  18432 

C,  =  18432  •^-  ^  =  4096 

Hence, 

(3+4:r')*: 

=729  +  5832a;'  +  19440x*  +  345602:'  +  34560a:' 

+  18432a:'-'' 4- 4096a:'«,^n*. 

(320) 

BINOMIAL  THEOREM.  219 


3      ;.     ^      *      1^ 
4,  n=i,     a=-,     6==^,     -=-. 


Ci 

=  a)* 

=  AV 

c. 

=  AV 

•fif=   fJ 

o. 

=  fj; 

•fif=   M 

c. 

=  ir 

•  1  •  II  =  Hi 

/7       1*2     •    1    •    il     ill 

^5     125         4  15  625 

Hence, 

/3a   4/-V   81  ,   27  ,   54  ,  ,   192  ,  256  .  , 
\4   5/   256    20     25     125     626  ' 


2,369 

n=6,     a=-,     o=~.     -  =  -. 

'  3'  2'     a     4 


0,  =  (!)• 

=  ^%^ 

c,  =  AV 

■f 

•f  =   H 

C,  =   M 

•f 

•  f  =   V 

C.  =   V 

'i 

■  J  =     20 

C,  =     20 

.  1  . 

4 

■  J   =  XJ» 

C.  =  xji  . 

•1 

•  f  =  n' 

C,    =    iLll  ■ 

i 

•  f  =  w 

Hence, 

/2<     3rY      64   .     32^,       20^,  ,     ^^^,  ,     135  ,  ,     243    . 


729 


1        A       1         *        * 

«=5,     «=j,     6=g,     -=-■. 


<7,  =  (J)' 

—  Tjir 

<?,  =  tAt 

•f 

•  t  =    Tk 

C»  =    ih 

•i 

•i=      T*T 

C,    =      TiiT 

•f 

•  J  =    5i,r 

0,  =    jh 

•r 

•}=  IH 

0,  =    ^h 

•i' 

•  1  =  jtVi 

Hence, 

U^sj  ~1024"256'^T60~200"^600~3T25  ' 
(320) 


220  SERIES. 

7.  n=8, 


Hence, 


.4. 

1 

'~2' 

b 
a 

=  1, 

c,  = 

a)' 

= 

il» 

c.  = 

lii- 

r 

1  = 

iV 

c,  = 

?v 

J 

= 

iv 

c,  = 

/i- 

1 
3 

= 

3V 

<^.  = 

/i- 

i 

= 

Wt 

c.  = 

t'sV- 

i 

= 

3't 

c,  = 

/J- 

1 

— 

j'l 

c.  = 

s'i- 

t 

= 

-A 

c.^ 

j'i- 

i 

^:^ 

Tk 

m'     1m' 

7m' 

+  • 

35 

7 

.+ 


1 


128     S27n*     64m*     32w* 

1 


256m' 


i,  ^n«. 


380,  page  323.) 


:V8  +  l  =  2Vl-|-|. 

^                             =  +1.0000000 

^=  +  ^  •|-^=  + 

416667 

C=-i-i-5  =  - 

17361 

i>=-i-|-C=  + 

1206 

JE=^l  .|-2)=- 

101 

^+  -HT^=  + 

9 

1. 0400420  =  Vl+| 

2 

V9  =      2.080084,  Ans. 

(320-323) 

DEVELOPMENT  OF   SURD   ROOTS.  221 


-/si  =V27  +  4=3^1  + _4^. 

A                              =  +1.0000000 

JB  =  +  J-  '^\'  A=z  +       493827 

C  =  -  ^  '^\'B  =  -        24387 

i>  =  -  I   •  aV  ^  =   +            2008 

^=~f     j\'J)=-             198 

^=~n-iV^=+        21 

1.047l27l  =  Vi 

+iV 

3 

V31  :=       3.141381,  Ans, 

Vl00=:v/i25  — 25  =  5^1  _j. 

^                              =  +1.0000000 

5=-i-l-^  =  -       666667 

C7=-i4-i?=-         44444 

i>=:-A.j.C7=-           4938 

^=-|-}-i>=-             658 

^=-|i*i-^=-               97 

G  =  -  I  'i'F=:^  —              15 

^=-if4-^=-             2 

0.9283l79  =  Vr 

-i 

5 

Vioo  =      4.641589,  Ans. 

VllO=Vi25-15  =  5<^l-^3^. 

A                               =  +1.0000000 

£  =  -^  '^\'A=  -      400000 

C=  —  |-^j-^=-         16000 

Drr— 1  .^j.e=--           1067 

^  =  --  f  •  /j  •  />  =  ~               85 

^=-H*i\-^=-             ^ 

6^=-^-jV-^=-                  1 

0.9582840  =  Vr 
5 

^^'V 

VllO  =       4.791420,  ^n<f. 

(323) 

222  SERIES. 

5.  ^297=^243 +  54=3^1+1. 

A  =  +1.0000000 

^  =  +  1  •  I  •  .1  =  +  444444 

C=-f  '%'  B  =  -  39506 

i>=-f-a-C=+  5268 

F—  -\\'\'  Ez=  +  138 

Q=  -L  'X'  F=  -  25 

/?=  -If' I*  6^=  +  * 


1.0409504 =Vn.  I 
3 


V297  =       3.122851,  Arts, 


V60=V64 

— 4i 

=  2Vl- 

-tV 

A 

=  +1.0000000 

B=  - 

-i   . 

tV 

.^ 

=  — 

104167 

C  =  - 

-^-.' 

tV 

■J? 

=  — 

2713 

I)=  - 

-u- 

tV 

(7 

=  — 

103 

E=- 

-u- 

I'a 

•i> 

=  — 

5 

0.9893012  =Vi_yy 
2 
Vei  =       1.978602,  Ans, 


7.  V4=V32-28  =  2Vl_i, 

In  this  case  the  last  term,  |,  is  nearly  equal  to  unity,  and  the 
series  will  converge  very  slowly.  We  may  give  the  calculation 
another  form  by  taking 

4  =  VaS  and  1/4  =  1^128. 

Whence,  |Vl28=|V243-115=^v/l_|i|. 

(323) 


DEVELOPMENT  OF   SURD   ROOTS.  223 


A 

= 

+  1.0000000 

B  = 

-  i- 

iH- 

A   = 

— 

946502 

C  = 

-tV 

iif  ■ 

B   = 

— 

179173 

D  = 

—,%■ 

Hi- 

C  = 

— 

50876 

E  = 

-a- 

iH' 

D  = 

— 

16854 

F  = 

-H' 

m 

'  E  = 

— 

6062 

G  = 

-H- 

m 

'  F  = 

~ 

2295 

H  = 

-If 

iii 

•  G  = 

— 

900 

I  = 

-a- 

iH 

'  H  = 

— 

362 

J  = 

-n 

•Hi 

'I   = 

— 

150 

K  = 

-u 

■Hi 

J  = 

— 

62 

L  = 

-a 

Hi 

'  K  = 

— 

26 

jr  = 

-If 

■Hi 

•  L  = 

— 

11 

N  = 

-If 

■Hi 

•  M  = 

— 

5 

0  = 

-li 

•Hi 

•ir  = 

( 

2 

}.8796720=v^l-iif 

f 

W  =  1.319508,  Ans. 


>^3275=V3125  +  150=6Vl+_|_. 

A  =  +1.0000000 

^  =  +  J  *  yf  5  •  ^  =  +        9^00^ 
C=  -f  -tIj-^  =  -  1843 

^  =  +  f  •  tIj  •  C'  =  4-  63 


1.0094208  =  ^^l+yf  3 
5 


v^3275  =       5.047104,  Ans, 
9.  v^T25=Vi28  — 3  =  2Vi_-.|^. 


A 

B  = 
0  = 
i>  = 

(323) 

+  1.0000000 

—  33482 

—  336 

—  5 

0.9966177  =  ^1- 

2 

-Tie 

1.993235,  Ans, 

224 


SERIES. 


Note. — A  table  of  logarithms  affords  an  easy  and  practical  method  of  finding 
the  roots  of  numbers,  since  we  have  only  to  divide  the  logarithm  of  the  number 
by  the  index  of  the  root,  and  look  out  the  number  corresponding.     See  40'ft. 

To  find  ^4,  we  have 

log.  4  =  0.6020600 

-^^  =  0.1204120 
o 

Hence,  V*  =  1-319608 

«8  in  Example  7. 


METHOD  OF  INDETERMINATE  COEFFICIENTS. 
(38a,  page  327.)  ^_^^ 


l-2a; 
l  +  dx 


:A-hBx+Cx*  +  I)x*+Ez*-^ 


0=     A 
—  1 


x'+  B 

x-¥    C 

x*+  B 

x^+  E 

-ZA 

^3B 

-3C7 

-32) 

+     2 

«*+. 


Therefore, 


whence, 


^-1=0, 

^=1; 

j5-3^  +  2  =  0, 

^=1; 

(7-3^=0, 

C=3; 

2)-3C=0, 

i>  =  9; 

JF-3i>  =  0, 

-£'=27,  etc. 

14-aJ  +  3x'  +  9j;'  +  27x*4-81a;*+ ,  Am, 


l4-2a; 
\—x—x 


^=A-\-Bx-^Ca^-\-Dx*  +  Ex*  + 


0=     A 

x'-hB  x-hC 

x*  +  D 

x*-hE 

—  1 

-A     -A 

-5 

-C 

-  2     -B 

-C 

-D 

0  -Aj  rr^f 


i  323-327) 

:% 


INDETERMINATE   COEFFICIENTS. 


225 


Therefore, 


■whence, 


A-l  =0, 
B—A-2  =0, 
C-'A-£=0, 


^=1; 

(7=4; 
/)=7; 

JE'=11;  etc. 


1  +3a;+4:r'4-Va;*  +  ll^*  +  18^»+ ,  Ans. 


Therefore, 


whence, 


1— a; 


l_3x— 2a;' 


■.A-{-Bx-\-Cx'-{-Dx*-i(-Ex*-^,, 


0=     ^ 
—  1 


x'-\-  B 
-^A 
+    1 


x+    C 
-2^ 


x'+  D 
-3(7 
-2J5 


a;'+  E 
-SB 
-2(7 


«*+. 


-4—    1=0, 

B-3A-{-    1=0, 

(7—35-2^=0, 

i)-3  (7-25=0, 

Jgr-3i>— 2(7=0, 


^  =  1; 
5=2; 
(7=8; 
i>=28; 
JS:=100;etc. 


l+2a;4-8a;'  +  28a:'  +  100a;*4-356a;»+ ,  Ans. 


J±^f^=A  +  Bx-\'Cx'+I)x*-{-Ex*-^ 
1— 4z  +  4a:' 


Therefore, 


"Whence, 


0=     A 
—  1 


x'-\-  5 
-4^ 
—  5 


x-h  C 
-45 
+  4^ 


x'4-  B  x'+   E 


-4(7 

+  45 


-42> 

+  4(7 


«•  + 


^-  1=0, 
5—4^—  6=0, 
(7-45  +  4^=0, 
2>-4(7+45=0, 
^-42) +  4  (7=0, 


.4  =  1; 
5=9; 
(7=32; 
i>=92; 
^=240;  etc, 


l+9a;  +32a;'  +  92a?'  +  240x*+ 

X 

«  +  9a;'  +  32a:'  +  92a:*  +  240a:*+ ,  Ans, 

(327) 


226 

SEB1E6. 

2 

■r                                           — 

.Ax-'  ^Bx'  -^  Cx^-Dx"  ^  Ex'  Jr , . . . 

*•              Zx-lx*- 

0=     3^ 
—     2 
Therefore, 

ar»  +  3^ 
-2^ 

3^- 

a;  +  3(7  ar'  +  Si) 
-2J?       -2C7 

2=0,             ^  = 

a;«  +  3jr  «*  + 
-22> 

35-2^=0, 

^=i; 

3(7-2J5=0, 

c=^S\ 

3i)-2C:=0, 

^=h; 

3^— 2i>=0, 

jg^= 

iVV,  etc. 

Whence, 


2      4     8a;     16a;'     32a;' 
8i  +  9  +  27  +  ^  +  2i3+---^'^- 


l+2a;'  +  3ap 


-^=Ax'-\'Bx-\-Cx*-\-Dx*  +  Ex'-\-   ... 


0=     A 

-  1 


Therefore, 


a;'+^ 

a;+    C 

a;'+  i> 

a;'+   E 

a;*+    i^ 

a;'^+    e' 

+  2  A 

+  2  J? 

+  2(7 

+  22) 

+  2^ 

+  3^ 

+  3^ 

+  3(7 

«•+ 


Whence, 


^-      1  =0, 

A= 

Ij 

B          =0, 

B= 

0; 

(7+2^=0, 

(7=- 

-  2; 

2) +2^=0, 

/)= 

0; 

3-4+^+2(7=0, 

E= 

1; 

3^+i^+2i)=0, 

F= 

0; 

8(7+(y  +  2^=0, 

G= 

4; 

3^+/+2(?=0, 

/=- 

-11; 

3Gr4.ir+2/=0, 

/{= 

10; 

3/+jlf+2^=0 

M= 

13,  etc. 

(7) 


l-2ar'  +  a;*  +  4a;«-lla;'  +  10a;"  +  13a;"— ,  Arts. 


Note. — It  is  obvious  that  that  the  alternate  equations  may  be  omitted ;  this 
is  done  from  the  sevenUi  onward. 

(327) 


INDETERMINATE   COEFFICIENTS. 
1  +  2ax  +  a  V 


227 


0=    A 
—  1 


x'+  B 
-{-2a  A 
—      1 


x+      C\x'+     D 
+  2aB\     +2aC 


x'+     E 

x*-\-     F 

-\-2aI> 

+  2a£ 

+  a'C 

+  a'J9 

x'+. 


Therefore, 


^-1=0, 

B-{-2aA  —  l=0, 

C+2aB  +  a'A=0, 

D'}-2aC-\-a^B=0, 

E-\-2aD->t  0^0=0, 


A=l; 

^=:-f(l-2a); 
(7  =  -(2a-3o'); 
i)  =  4-(3a'-4a^); 
^=-(4a'  — 5a*);  etc. 


Whence, 
1  +(l-2a)a:-(2a-3a>'  +  (3a'-4a')a;'-(4tt'-5a>*+  . . . ., 

Ans, 


8.  \r[:^=A-\-Bx-\-Cx''-\-Dx'-\-Ex'  + 

Squaring  both  sides,  we  have 


l-x^zA'  +  AB 
+AB 


X  +AC 
-hAC 


Therefore, 


A'-l  =  0, 
2AB-\-l  =  0, 


2AC-{-B*  =  0, 

2AI)-\-2BC  =  0, 

2^^+C"  +  2^i>  =  0, 


Hence, 


x'  +  AD 
+  BC 
+  BC 
+  AJ) 


x'  +  AE 
+  BD 
+  C' 
-hBD 
+  AE 


^-2-4' 


E  = 


x'-^,. 


2-4-6' 

3-5 
2-4-6-8 


;  etc. 


X      x* 


ri-«=l-;T-;r^- 


Zx"" 


^■bx* 


2     2-4     2-4-6     2-4-6'8 
(327) 


. . . ,  Ans* 


228 


SERIES. 


9.  Assuming  the  same  form  of  development  as  in  Example  8,  we 
shall  have, 

^'-1  =  0,  ^  =  1; 

2AB-S  =  0,  -5  =  1; 

2AC+B'-5  =  0,  C7  =  V; 

2AD  +  2BC-1  =  0,  ^  =  H; 

2AE+C-{-2BI)-9  =  0,  H  =  HI ;  etc. 


"Whence, 


,      Sx     liar'  .  23x'  .  Il9x*  . 

2        8         16        128  ' 


10. 


— ; ? — i i— ^ rs =  Ax'-\-Bx'-^  Cx'  +  Dx*-\- 

l+a;'-fa:*  +  x«  +  a:'4-a:'°+ 


0=     A 
-1 


x'  +  B 

■\-A 
+  2 


+  A 
—3 


a:«  +  i> 

+  A 
+  4 


a:«  +  ^ 
+  i> 
+  C 
-¥B 
+  A 
—  5 


x*-\-F 


Whence, 


^-1  =  0, 

B+A-^2  =  0, 

C+B  +  A-3  =  0, 

i)  +  CH-^  +  ^  +  4  =  0, 

l—dx*+5x*—7x'+9x'—lW  + 


+  E 
+  i> 

+  (7 

4-6 
^=      1; 
^  =  -3; 
C  =  +5 ; 
2>  =  —7  ;  etc. 


r"  +  ..., 


.,  Arts. 


REVERSION  OF  SERIES. 
(383,    page  330.) 
1.  Here  we  have  0=1,   i=l,   c=:l,  etc.;  whence  by  formula  (J^), 


Hence, 


^=1,    ^=-1     C=l,     D=-\,    E=\,  etc. 

*=y— y'+y'— y*+y'— — »  ^'i** 

(328-330) 


.  .  etc. 

5 


REVERSION   OF  SERIES.  229: 

2.  a=l,     6=3,     c=5,  etc. 
By  formula  (T), 

A=zl,     B=-3,     (7=13,     i)=— 67,     -^=381,  etc. 
Whence, 

«=y-3i^'  +  13y'-67/  +  381/-....,  Ans. 

3.  a=l,     6=-^,     c=i,     rf=-i,     c=i,  etc. 
By  formula  {F)y 

A=l,    B=-,     C=—,    i>=— i_,     B=z , 

'  2'  2-3'  2-3-4*  2-3-4-5' 

Whence, 


4.  a=l,     6=  — 1,     c=l,     c/=— 1,     e=l,  etc. 
By  formula  (G), 

A  =  l,     B=l,     C=2,     i>=5,     J^=14,  etc. 
whence, 

ar=y  +  y'  +  2y*  +  5/  +  14y'+....,^n«. 

5.  a=2,     6=3,     c=4,     rf=5,     «  =  6,  etc. 

By  formula  (6^), 

A=i,     B=-j\,     C=V/3,     i>=-TV/4,etc. 
Whence, 

y     3/19/     152/ 
''-2""l6"^T28  ""1024  +  •  •  •  "  ^''^• 

6.  a=2,     6=4,     <'=6,     c?=8,     c=10,  etc. 
By  formula  (F), 

A=^,     B=z-i,     C=f,     D=-h     ^=Hf  etc. 
"whence, 

_x     x*     5x'     7a;*     21a;" 
^""2""^    T"    "s"    le" 
(830) 


230  SEKIES. 

(384,  page  331.) 
In  the  following  examples,  let  s  be  the  sum  of  the  series. 

1.  By  formula  (F), 

1        4,2-8,      4-16  ,      8-32    , 
6       25    ^125     ^625       ^3125 

Substituting  the  value  of  «=}, 

^=1(1)*+ (!r+2(i)'+4-.(i)'+8  •«)■"+.... 

This  is  a  geometrical  series  in  which  r=2(f)';  and  by  361, 

x=^  ^^^;  ,,=0.11764706  +  ,  Ans, 
1-2(1)' 

2.  By  formula  (F)y  we  have 


f  ~3«*'  ""^To-*  "■alVo'" 


Substituting  the  value  of  *=|,  we  find 

«  =  -f  0.500000 

~|s»= -0.041667 

— ^V«'=-0'003472 

.  -^|^«*= -0.000231 

__ji^_5»=, -0.000010 

Therefore,  ~  ar=  4- 0.454620,  ^>w. 

3.  By  formula  (G),  we  have 

Hence, 

«  =+0.200000 
+  i5»= +0.001333 

Therefore,  '  *  «r=     0.201369,  Ans. 

(331) 


RECURRING   SERIES. 
4.  By  formula  (F), 

x=s  4- !«'  +  ^V*'  +  4^8**  +  3  if  f  o«*  + 
Hence, 

s  =+0.250000 

|«»= +0.023438 

^^«'= +0.001139 

j|i^5*= +0.000074 

9lffo«'= +0.000004 


Therefore, 


ar=       .274665,  Am. 


Mi 


RECURRING   SERIES. 

(391,  page  335.) 
1.  We  have  by  formula  (P), 


whence, 

By  formula  (g), 


m  +  Sn=4, 
8m  +  4n=1 ; 
m=l, 
n=l. 


1+3^;— ar       l  +  2a; 

o=- =- ,  Am, 

l—x—x*     1— ar— ar'* 


2.  By  formula  (P),  we  have 

6m  +  12»=48; 
m—  6, 
n=  1. 


whence, 

By  formula  (§), 


l  +  6.r— a;  _    l+5a; 
l_a:-6a;'~"l-a;-.6a; 


^ ,  -^w*. 


(331—385^ 


23*2  SERIES. 

3.  By  formula  (P),  we  have 

m  +  2n=—  5, 

2;/i  — 5n=4-26; 

whence,  w=  +   3, 

»=—  4. 
Hence  by  (^), 

l4-2ar-f-4ar_       l4-6a; 


4.  By  formula  {T)t  wo  have 

m  +  4n  +  3r=—  2, 

4wi  +  3»  — 2r=+  4, 

3m— 2»  +  4r=  +  l7; 

whence,  wi=  +  3, 

n=—  2, 

r=+  1. 
Hence  by  ( F), 

l-jr  +  2ar'-3x'  l-ar4-2ar'-3«' * 


6.  By  formula  (P),  we  have 

wi4-3w=     5, 

3m +  5/1=     7; 

whence,  m=  —  1, 

n=+2. 

By  (C), 

„     l4-3i;--2a;       1+ir 

6.  By  formula  (P),  we  have 

wi4-w=  5, . 
m  +  5n=13 ; 
whence,  wi=  3, 

n=  2. 

By  (C), 

^_  l+g— 2g    _         1—3?  . 

^"-l^2x-dx'''l--2x^Zx**  "^""^ 
(335) 


RECURRING   SERIES.  233 


7.  By  formula  (T),  we  have 

m  +  4n  +  6r=  11, 

4m  +  6n  +  llr=   28, 

6m+llw  +  28r=  63; 

whence,  m=-f-3, 

n=-l, 
r=+2. 


By(r). 


l-f4a;  +  6a;'  — (l-f  4a:)-2a;+a;* 
1  — 2a:  +  a;'— 3a;' 


_l4-2a;-2j:'  +  ig'_(l+a?)'-2.g* 
l_2x  +  a;'-3a;»~(r^a:)'-3«' ' 


8. 
lei 

By 

ice, 

formula 

(n 

we 

have 
2 

7n_ 

■  2  ~ 

m= 

n= 

1 
2' 

10; 

3, 
2. 

By  (§)  we  may  now  sum  the  series,  but  we  must  notice  that 
equations  (2),  (389),  will  now  have  the  form, 

I>=mBx*  +  nCx^^  etc.; 
or  that  we  must  change  a;  in  (Q)  to  x'.     Hence, 

-+X*'-X' 

„_  2  _  X 

^-l-2x'^dx-2-4x^-6x*'  "***'• 


(335) 


234  SEBIES. 


DIFFERENTIAL  METHOD. 
(395,  page  339.) 

1.  a  =  l,     rf,=3,     c?3=l,     d^  =  0. 
By  formula  (A), 

8*7 
r,  =  l+3*8H — —  =  53,  ^n«. 

2.  a  =  l,     rf,=3,     rfa=3,     rf^^l,     rf^=0. 
By  formula  (A)^ 

3.  a=l,     c?,=5,     rfj=10,     ^3  =  10,     c?^=5,     dj.=0. 

Hence,  ^s^'^^l,  and  ^^  =  1231,  Atis. 

4.  a=l,     rf,=7,     ^3=12,     ^3  =  6,     rf,=0. 

r,,=l  +  7'19  +  ^^-12  +  ^^'\^'^^-6=:8000,^y... 
2  2    3 

5.  a=l,    rf,=2,     ^3=1,     c?3=0. 

2  +  4n— 4+n'  — 3rH-2     n(n4-l)      . 

= =  -^ — - ,  Ans, 

2  2       *  , 

(339) 


DIFFERENTIAL    METHOD.  235 

a=l,     rf,=.3,     d^=S,     c?3  =  l,     d^=zO. 

m      ,      «/       ,^     (n  —  l)(n-2)          (n-l)(n  —  2)(n^3) 
T,=  l+3(/i~l)  +  '^ ^^ -^-3  +  ^ 2^"^ 


6 
n*  +  3n'  +  2n_n(n  +  l)(n  +  2) 


6  6 


,  Ans. 


7.  a  =  l,     e?,=4,     «?a=6,     d^=4,     d^=l. 

^            w       ,v      (n—l)(n—2)    ^      (w— l)(w-2)(n~3)    , 
7;=l+4(n-l)  +  ^ ^ ^'6  +  ^ 2-3    "^^'*" 

{n-l){n-2){n-S)(n-4) 
2-3-4 

__            _      n*4-6n'  +  ll»'4-6n     n(n4-3)(n'4-3»4-2) 
Or,  T^= 24 = 24 

n{n  +  S)(n  +  2){n  +  l) 
= 24 '  ^    • 


8.  a=l,     c?,=2,     rf2=l,     <f3=0. 

By  formula  (i?),  (394), 


«     «^     20  •  19     ^     20  •  19  •  18      ,^^^      . 

5=20  +  — - — -2+ — — =1540,  Ans. 

2  2  *  u 


9  a=l,     rf,=4,     ^3=5,     ^3  =  2,     rf^=0. 

^     ,^     12-11      ,      12  •  11  •  10     ^12 -11  -10 -9     „ 
5^12  +  __.  44— 2T3— •  5  4—yTyT-r-  •  2; 

or  5=2366,  Ans. 


10.  a=l,     rf,=3,     ^3=6,     d^  =  9,     d^=0. 

10-9  10  •9-8     ^10-9-8-7     ^ 

5=10  +  ^-  .  3  +  -2T3-  •  6  +  -2T3T-r- •  9; 

or  5=2755,  Ans. 

(339) 


236  SERIES. 

11.  a=2,     <f,=4,     ^3=2,     ^3=0. 


r*     «     .  w(n  — 1)     ,      n(n-l)(n— 2)     ^ 
^=2n4--^--2— ^  •  A+-^—^ ^  •  2 ;  or 

^^n'  +  3;i'  +  2^4n4-2)(n4-l)   ^^^ 
3  3  *         ' 


12.  a  =  6,     </,=18,     rfj,=18,     <f3  =  6. 

o     ^       »(«-l)    ,o     n(n-l)(n-2)  ,        n(n-l)(n-2)(n-3)   ^ 

^_n'  +  6n*4-lln'  +  6»     n(n  +  3)(n  +  2){n  +  l) 

^— ^1^ = ,  Ans. 


13.  The  series  in  examples  13,  14,  and  15  may  be  summed  by 
formula  (B)^  (389) ;  but  the  following  method  by  indeterminate 
coefficients  is  easier. 

Assume 

l'  +  2'+3'+ +n*  =  A  +  Bn+Cn*-\-Dn'i-JSn*+ (l) 

Change  n  into  n  4- 1  ;  then 

l«  +  2'  +  3«+....+n'  +  (n  +  l)«=^  +  ^(n  +  l)+C(»  +  l)' 

+  i)(n  +  l)'  +  ^(n  +  l)*+....  (2) 

By.  subtraction, 

n'+2n  +  l=B+C{2n-{-l)  +  I>(Sn*  +  Sn  +  l) 

+  E(4n'  +  6n^  +  4n  +  l)  + 

or,  by  arranging  terms  with  reference  to  the  powers  of  n, 

»'  +  2»  +  l=4^»'  +  (3i>  +  6^)n'  +  (2C7+3i)  +  4^)n 

+  (5+C7+i>  +  ^)  +  ....  (3) 

Now  by  (368,  III.),  we  have 

iF=0,  or  JS:=0; 
and  the  same  value  for  all  coefficients  beyond  JS, 

Equating  coefficients  of  like  powers  of  n  in  (3),  omitting  the 
terms  containing  JE',  we  have  the  following  equations  of  condi- 
tion: ^  " 

(339-340) 


DIFFERENTIAL   METHOD.  237 

3i>=l,  whence  i>=|; 

2(7+3i>  =  2,  "  (7=1; 

i?+C7+i>=l,  "  jB=i. 

And  by  substitution  in  (l), 

l.  +  2'  +  3'+....+„'=^  +  ^  +  |'  +  J, 
To  determine  A,  put  n=l  ;  then  -4=0,  and  we  have 

^=6  +  2  +F= 6 = 6 ■ '  ^"'- 

In    Example   14,  we  shall   have   ^=0,  and  the  same  for  every 
coeflScient  beyond  F.     The  equations  of  condition  arc 

4ii'=:l,     whence     -fi^=i ; 


3Z>  +  6^=3,  "  D= 


a  1 
2C'4-3i>  +  4jE'=3,  "  0  =  1'^ 

Hence, 

^     n*     n'     n'     (^'-fn)'     . 
^=T+2+4=^-T^»^"" 

In  Example  15,  we  have  G=0,  and  the  same  for  every  coeflScient 
beyond  G.    The  equations  of  condition  are 

5F  =  1,  whence  /"=}, 

4^+10/^=4,  ''  ^=1, 

8i)  +  6J^+10i^  =  6,  "  i>=|, 

2^(7+ 32) +  4^+   5i^=4,  "  (7  =  0, 

^+(7+i>  +  ^+/'  =  l,         "  J?=-^V. 

Hence,       .  8=.-  +  -  +  - --,  Ans. 

In  this  solution  the  symmetry  of  the  equations  that  determine 
Fj  F,  Dy  «fec.,  should  be  noticed.  Taking  the  numbers  in  vertical 
columns,  they  are  the  binomial  coeflScients.  We  can  easily  extend 
this  method  to  the  fifth  and  sixth  powers ;  and  so  on. 

(840) 


238 


SERIES. 


16.  Performing  the  multiplications  indicated,  we  shall  have  the 
two  series, 

m(l+2  +  3+ +n) 

and,  l'  +  2'  +  3'+....+7i'. 


Hence, 


or, 


-,     «(»  +  !)         n(n  +  l)(2n+l)^ 


INTERPOLATION. 


(397,  page  341.) 

For  the  first,  second,  and  third  examples,  we  have 
a=2.V58924,     rf,  = +.043115,    rf^  =  -.001287,    ^3= +.000091. 

For  the  fourth,  fifth,  and  sixth, 
a=2.802039,     rf,  = +.041828,      e?^  =.001196,     (/,= +.000083. 

By  substitution  in  the  formula,  we  obtain  the  following  results ; 

1.     1st  term,  +2.758924  2.     1st  term,  +2.758924 

2d      "      +      14012  2d      "      +      37726 

3d      "      +           141  3d      "      +             70 

4th     "      +               6  4th     "+               2 


2.773083,  Ant. 


2.796722,  An\ 


Isttcrm,  +2.758924 
2d  "  +  19695 
3d     "       +  160 

4th    "       +  6 


2.778785,  Ans, 


1st  term,  +2.802039 
2d  "  +  10457 
3d     "       +  112 

4th    «       +  6 


2.812613,  Arts, 


5.  Isttei-m,  +2.802039 
2d  "  +  28611 
3d     "       +  129 

4th   «       +  4 


2.820783,  Ans. 


6.  1st  term,  +2.802039 
2d  "  +  31371 
3d     "       +  112 

4th    "       +  3 


(340-341) 


2.833525,  Ans. 


INTERPOLATION. 


23Sr 


(398,  page  342.) 


FUNCTION. 

d. 

d. 

d. 

66°      6'    38" 

72     24        5 

+  6°   17' 

27" 

Y8     34     48 

+  6     10 

43 

-6'    44" 

■ 

84     39        4 

+  6       4 

16 

-6     27 

+  17" 

90     37     18 

+  5     68 

14 

-6       2 

+  25 

96     29     57 

+  5     62 

39 

—  5     35 

+  27 

Note. — In'  forming  the  differences  we  may  use  as  a  check  to  guard  against 
numerical  errors,  the  rule,  "  The  sum  of  the  differences  in  any  column,  plus  the 
first  term  of  the  preceding  column,  is  equal  to  the  last  term  of  the  preceding 
column." 

Observe  that  for  the  1st,  2d,  and  3d  examples, 
.    a=66°6'38';    rf,  = +  6**  17' 27';    d^  =  '-6'W\    d^=+ir\ 
For  the  4th,  5th,  and  6th, 

a  =  72°24'5",    rf,  = +6®  10'43",    c?3-6'27",    fl?3  =  +  25". 
For  the  7th,  8th,  and  9th, 

a  =  78°34'48",    rf,  = +6°  4' 16",    d^  =  -^Q' 2'\    (£3  = +  27". 
Again,  we  have 

for  examples  1st,  4th,  and  7th,  **=?  j 

"  "  2d,  5th,  and  8th,  n=^ ; 

•*  "  3d,  6th,  and  9th,  n=^. 


1.    1st  terra,  +66°   6' 38' 


2.    1st  term, +66°   6' 38" 


2d     " 

+    1   34  22 

2d     " 

+    38  43.5 

3d     " 

+               38     . 

3d     " 

+              50.5 

4th   " 

+                 1 

4th    " 

+                1 

67°  41' 39",  ^n5. 

-   69°  16' 13", 

(342) 

240  SERIES. 

3.    1st  term, +66*   6' 38"  4.    1st  term, +720  24'    6" 

2d     "+    4  43     5  2d     "       +    1   32  41 

3d     "       +              38  3d     "       +              36 

4th   "       +                1  4th    "       +                1 


10^*50'  22",  Ans,  73^67'  23",  Ans, 


6.    1st  term, +72°  24'    5"  6.    1st  term, +72°  24'    5' 

2d     "       +35  21.5  2d     "       +    4  38     2 

3d     "       +              48.4  3d     "       +              36 

4th    "       +                1  4th    «       +                1 


75**  30'  16",  Ans.  77°   2'  44",  Ans. 


7.    1st  term, +78°  34' 48"  8.    1st  term, +78°  34' 48" 
2d     "+    1   31     4  2d     "       +    3     2     8 

3d     "       +  34  3d     «       +  45 

4th   "       +  1  4th    "      +  2 

80°   6'27",^rw.  81°37' 43",^iW. 


9.    1st  term,  +78°  34'  48" 
2d     "        +4    33    12 
3d     "        +  34 

4th   "        +  1 


83°    8'  85",  Ans. 


LOGARITHMS. 
(416,  page  355.) 

2.      log.  104=2.017033  3.        log.  73  =  1.863323 

"         5=0.698970  "       2  =  0.301030 


2.716003,  Ans.  2.164353,  Ans, 

(842-356) 


LOGARITHMS. 


241 


4.        log.  50  =  1.698970 

«     29  =  1.462398 


5.       log.  .53  = 
3  = 


3.161368,  Ans. 


-1.724276 
0.477121 


0.201397,  Ans. 


6.     log.  1017  =  3.007321 
"  2=0.301030 


7.  log.  10.91  =  1.037825 
"  7=0.845098 


3.308351,  Ans, 

1.882923 

8. 

log. 
u 

.01005: 
2: 

=  -2.002166 
=  0.301030 

9.   log. 
ns. 

.91  = 
.42  = 

-1.959041 
—  1.623249 

2.303196,  A^ 

—  1.582290, 

Ans, 

10 

1, 

log.  103  = 
«   15  = 
"   11  = 

=  2.012837 
rl. 176091 
.1.041393 

4.230321 

(41T,  page  356.) 


log.  10720  = 
405  X  0.4  = 


4.030195 
162 


4.030357,  Ans. 

log.  10.85 

=  1.035430 

400  X  0.39 

=     156 

1.035586,  Ans. 

log.  1021 

=  3.009026 

425  X  0.56 

=     238 

3.009264,  Ans, 

log.  5.6 

=  0.748188  . 

:.  101.5232 

=  2.006565 

3.754753,  Ans, 
(365-356) 


242  SERIES. 

1.  log.  3  =  0.477121 

log.  1081.333  =  3.033960 


3.511081,  Ans. 


8.  log.  3.6  =  0.556303 
log.  101.4601  =  2.006295 

2.562598,  Ans, 

9.  log.  1.3  =  0.113943 
log.  101.977  =  2.008502 

2.122445,  Ans, 

10.  log.  5.6  =  0.748188 
log.  101343.04  =  5.005794 

5.754982,  Ans. 

11.  log.  2.5  =  0.397940 
log.  103.48  =  2.014856 

2.412796,  Ans, 

12.  log.  1.2  ==  0.079181 
log.  1.08  =  0.033424 

0.112605,  Ans. 

13.  log.  57  =  1.755875 
log.  101.4737  =  2.006353 

3.762228,  Ans, 


(356) 


LOGARITHMS.  243 


.     EXPONENTIAL   EQUATIONS. 
(418,  page  357.) 


1.  Taking  the  logarithms, 
X  log.  7  =  log.  8 


locr.  8     0.C03090     , 
-— ^— — —1.06862,  Ans, 


log.  7     0.845098 


2.  Taking  the  logarithms, 
-(log.  5)  =  log.  30. 

X 


2  log.  5     1.397940     ^^,,„,,     , 
a^=i — ^  =  7-77^777777=0-946395,  Ans. 
log.  30     1.477121  ' 


3.  X  log.  a  — 2  log.  6  +  3  log.  c 

2  loff.  6  +  3  logr.  c      ^ 

•whence,  ir= 2_j 2 — ,  ^w*. 

loor.  a 


4.  Clearing  of  fractions  and  transposing,  we  have 

ah' :=.din -\- c 
log.  a  +  ar  log.  6  =  log.  (c?w  +  c) 

loff.  (c?m  +  c)  —  log.  a      . 

whence,  x^-^-^ — -, '- ^—  ,  Ant, 

log.  6 

5.  log.  m  +  -{log.  a)  =  log.  6, 

whence,  x—-, j^^ ,  Ans, 

log.  6— log.  m 


6.  Adding  and  subtracting  the  equations, 

a'=:c  +  (/,  and6''=c— 6?; 

log.  (<:  +  c?)  log.  (c— c?) 

whence,  a; = -V^ -,  y=    ^,  ^    ,— ^ ,  -/^ w5. 

log.  «  log.  6 

(357-368) 


244 


PROPERTIES   OF   EQUATIONS. 


V.  Taking  the  logarithms, 


8. 

whence, 

9. 

whence, 


_log.  729_log.  3'_6  log.  3_ 
log.  3       log.  3        log.  3        ' 


(log.  216)  =  log.  12, 

_3  log.  216_3  log.  6'_9  log.  6      . 
'^~    log.  12       1^12"~log.l2  '         ' 


(log.  616)=log.  12; 


__3  log.  516 _3  log.  43  +  3  log.  12 


log.  12 


log.  12 


3  log.  43      „     , 
-        »— +3,^n,. 


log. 


10.  Taking  the  logarithms, 

X  log.  6  =  6  log.  24  +  ^  log.  17  — log.  71 

18log.24  +  loor.  17  — 3loff.7l       , 

whence,  x= — r^ — ,  Am. 

^  3  log.  6  ' 


PROPERTIES  OF  EQUATIONS. 
(428,  page  366.) 

1.  Factors,      <        ^     ^ 
Product,  a;'— X— 0  =  0,  Ans. 

2.  By  (4^3),         the  1st  term  is     ar* ; 

"     (434,   1)       "    2d       "        6a?»; 
"     (434,  2)      "    3d      "         2x; 
"     (434,  5)       "    4th     "         -8. 
Hence,  a:'  +  5i;'  +  2.c— 8  =  0, -4n*. 

(358-366) 


PROPERTIES   IN   EQUATIONS.  240 


Factors, 


ar— 3=0 

ic  +  2=0 
a;+l=0 
x  +  5  =  0 


Product,  ar*  — 5x'  — Tar' +  29^  +  30=0,  ^n«. 


Factors, 


ar-(l+f/--5)=:0  (1) 

;r_(l_|/_5)=0  (2) 

X-V5  =0  (3) 

x  +  V^S  =0  (4) 

Product  of  (1)  and  (2),     x*—2x  =6  (5) 

(3)     -    (4),      x'-5  =0  (6) 

"  (5)     "     (6),      x*-22r»  +  a;'  +  10a;-30=0,  ^ns. 

_    5.                     By  (423,)  The  1st  term  is  «'; 

«     (4a4,   1),      "     2d         "  -4x*; 

**     (4^4,  2),      "     8<i         "  Ox'] 

"     (404,  3),      "     4th        "  22a;'; 

«     (424,  4),      "     5th        "  -26a:; 

"     (4«4,   5),      "     6th        »  -42. 

Hence,  the  required  equation  is 

a;>  — 4a:*  +  22a:'  — 25jr— 42  =  0,  Ans, 

Or  thus : 

ar+l=0  '    (\/ 

ar  +  2=0  (2) 

Factors,      }                   ^3=0  (3) 

;r_(2+^-3)=0  (4) 

x-{2'-i^s)=0  (5) 

Product  of  (4)  and  (5),  a:' -4a; +  7=0. 

Hence, 

(x'-4ar  +  7)(ar  +  l)(.r  +  2)(a;-3)=a;»-4.r*±0a;'  +  22a;'-25ar-42=0, 

Ans, 

(366-367) 


246  PROPERTIES  IN   EQUATIONS. 

6.  {x'^5x^  +  ldx—2l)-r-(x-3)=x^-2x-{-1=0j  Ans, 

1,  (x*-\-2x'—34x*  +  12x-{-35)-^(x-\-1)=x'-5x*-\-x  +  5=z0j  Ans, 

8.  (x  —  2){x-\-3)=zx*  +  x—6',hencey 

(x*-3x'—4x^-\-30x-3Q)-T-{x^+x-Q)=x*'-4x+6=0, 

the  depressed  equation,  which  may  be  solved  thus : 
x'—4x=-6, 

ic=2+*^^,  or  2—^^, 


COMMENSURABLE   ROOTS. 

(432,  page  373.) 
2. 
Divisors,  6,         3,         2,         1,       —1,       —2,       —3,       —6. 

-6 
Quotients,     —1,     —2,     —3,     —6,  6,  3,  2,  1. 

Add  11 

10,         9,         8,         5,         17,         14,  13,         12. 

2d  quotients,  3,  4,  5,      —17,        —7,  —2. 

Add  -6 

—  3,      -2,      -1,      —23,      —13,  -8. 

3d  quotients,  —1,     —1,     —1,         23, 

There  are  three  final  quotients  equal  to  —  1  ;  and  the  correspond- 
ing divisors  are  3,  2,  and  1.  Hence  the  given  equation  has  three 
commensurable  roots,  1,  2,  and  3,  Ans, 

(367-373) 


COMMENSURABLE   ROOTS. 


247 


O        P        j^ 


H 


H4»      »^ 


i  ^ 
i  I 


§  - 


S" 

89 

J* 

p 

^ 

in- 

e^ 

1 

1 

O 

J-» 

o 

■ 

o 

1 

JO 

(ft 

I 

1 

p 

1 

CO 

aq 

-• 

Bi 

e 

<' 

p 

f 

e- 

§ 

g 

^ 

«r»- 

S9 

CD 

o 

rt) 

to 

►o 

" 

P 

P 

1 

— '. 

O* 

>• 

P 

»-!• 

1 

CO 

1 

O 

JO 

P^ 

so 

*< 

p 

o 

a- 

-^ 

1^ 

1 

CO 

p 
o 

EJ". 
O* 

P 


|>      CO 
Cl,   ^ 

p 
o 


P 
o 


o 


>  c> 


I    I 

^^     to 


+  I    I 

(—1 
rfi.     J35     JO 

I  I 

CO     «o 


03       CO 


to    jf^ 


H-l        CO 


I   I 

h-t      CO      CX) 


p 


g 

55' 
o 


1 

1       1 

OS     J-" 

00 

1 

JO 

1 

1 

1 

►o 

1 

1 

lO 
JO 

1 

J3S 

1 

to 

CD 

1 
JO 

1 

I—" 

to 

1 

t— • 
JD 

J35> 

1 
JO 

4^ 

1 

J^ 

1 

to 

en 

t—i 

(373) 


248  PROPERTIES  OF   EQUATIONS, 

4. 

Divisors,          21,  7,         3,         1,       —1,       —3,       —7,    —21. 
21 

Quotients,          1,  3,         7,       21,     —21,       -7,       -3,       — 1. 
Add              -16 

—  15,  —13,     -9,         5,     -37,     -23,     —19,     —17. 

2d  quotients,  —3,         5,         37. 

Add  -6 


—  9,     -1,         31, 
3d  quotients,  —3,     —1,     —31. 

Add  ±0 


—3,     —1,     -31. 
4th  quotients,  —1,     —1,         31. 

The  final  quotients  show  that  the  given  equation  has  two  com* 
mensurable  roots,  3,  and  1,  Ant. 


6. 

Divisors,  10,         5,         2,  1,       -1,     -2,     -5,     -10. 

—  10 

Quotients,       —1,     —2,     -5,-10,         10,         5,         2,  1. 

Add  2 

1,         0,     -3,       -8,         12,         7,         4,  3. 

2d  quotients,  0,  —8,-12. 

Add  5 


5,  -3,       -7. 

3d  quotients,  1,  — 3,  7. 

Add  -6 


-5,  -9,  1. 

4th  quotients,  —1,  —9,       —  1. 

(373) 


EQUAL  ROOTS.  249 

The  final  quotients  show  that  5  and  —1  are  the  commensurable 
roots  of  the  given  equation.     Hence, 

x-b=Q,  (1) 

ir  +  l=0.  (2) 

Dividing  the  given  equation  by  the  product  of  (1)  and  (2),  we 
have 

a;'— 22'+l  =  -l, 
Therefore  the  four  roots  are  5,  —1,  1  ■\-V~^\^  and  1— i^^,  Arts, 


EQUAL  ROOTS. 

(435,    page  379.) 
2.  We  have  given 

The  first  derived  polynomial  of  the  first  member  is 

X,=5a:*  +  8x'  — 33a:'-16a;  +  20. 

The  greatest  common  divisor  of  this,  and  the  first  member  of  the 
given  equation  found  by  (1045),  is 

D=x'—x-1  =  {x-2){x  +  \), 

Therefore  x—2  is  twice  a  root  of  the  equation,  also  x=—\  is 
twice  a  root ;  and  the  equation  has  two  roots,  each  equal  to  2  ;  and 
two  roots,  each  equal  to  —1. 

Dividing  a;'  +  2.c*-lU'-8j:''  +  20.r  +  16=0,  by  (:r-2)'(«  +  l)'=:0, 
we  have  ar  +  4  =  0,  and  x—  —  i.  Hence  the  roots  of  the  given  equa- 
tion are,  2,  2,  —1,  —1,  —4. 

(377-379) 


250  PROPERTIES  OF   EQUATIONS. 

3.  We  have  given 

whence, 

X,  =5x*-8j;»  +  9j;'— 14a;+8=:0. 

By  (105)  we  obtain 

D=x^—2x-{-l  =  {x-iy=0. 

Therefore,  since  x=l   is  twice  a  root  of  i>,  by  (4:3«5,  II)  it  is 
three  times  a  root  of  the  given  equation  X. 

4.  X  =  x*  —  2x'—Ux*  +  l2x  +  S6=0; 
X^=4x'-6x^-22x  +  l2  =  0. 

Whence,  by  (105), 

D=x^-x—6  =  {x—S){x  +  2)=0, 

Hence  x=3  is  twice  a  root  of  X,  and  also  x=  —  2  ;  and  as  the 
equation  is  of  the  fourth  degree,  it  has  only  four  roots.     Therefore, 

3,  3,  —2,  —2,  Ans. 


TRANSFORMATIONS. 
(437,  page  384.) 

5.  Put  x=y-\-2]  then  x'=2.     Whence, 
X'   =(2)*-4(2)»-8(2)  +  32, 
X,=  4(2)'-12(2)'-8, 

X\_  3-4(2)'     2-12(2) 

2    ~        2  2       ' 

X3_  2-3-4(2)     2-12 
2-3~       2-3  2-3  ' 

X,   _  2-3-4 
2-3-4~' 2-3-4'  "     2-3-4 

Therefore  the  transformed  equation  must  be 

y  +  4/  — 23y=0,  Am, 

(379-384) 


or 

X  = 

0; 

or 

X,  =  - 

-24; 

or 

2 

0; 

or 

2-3 

4; 

or 

^4_ 

1. 

DETACHED   COEFFICIENTS.  251 

6.  Putar=y— 3;  thcna;'  =  — 3.     Whence 
X'  =:(-3)*  +  16(-3)='  +  99(-3)'  +  228(-3) 

+  144,  or     X'   =       0; 

-X",=4(-3)'  +  48(-3)'  +  198(-3)  +  228,  or     X',=:-42; 

-X-,_3-4(-3r     2.48(-3)     198  ^^     X',_ 

X^3_2-3-4(-3)     2-48  X'3  _ 

2-3             2-3         ■^2-3*  ^^      2.3~        ^' 

X\        2-3-4  X'. 


2-3-4"2-3-4'  ^^2.3.4-        ■^' 

Therefore,  the  transformed  equation  must  be 

y*  +  4y'  +  9y'  —  42y  =  0,  Ans, 


v.  By  (437),  put  a:=y  +  |,  then 

x'-. 

=2.     Whence, 

X'   =(2)*-8(2)'4-(2)'  +  82(2)- 

60 

=     60; 

X\=4(2)'-24(2)»+2(2)  +  82 

=     22; 

X\      3-4(2)'     2-24(2)     2 
2             2                2       "'"2 

=  -23; 

X'3      2-3-4(2)     2-24 
2-3-      2-3           2-3 

=       0; 

X',        2-3-4 

=        1. 

2.^.4      2-3-4 

Therefore,  the  transformed  equation  must  be 

y*  — 23y'  +  22y  +  60  =  0,  Arts. 

MULTIPLICATION  AND  DIVISION  BY  DETACHED 
COEFFICIENTS. 


( 

44O5 

page  390. ) 

3-2-1 

5. 

3-5-10 

4  +  2 

2-4 

12  —  8-4 

6—10-20 

_,-6-4-2 

-12  +  20  +  40 

12-2-8-2 

6  — 22d=   0  +  40 

(38£ 

1-391 ) 

252 

PROPERTIES 

OF 

EQUATIONS. 

6. 

1+1+1 

1. 

l_44-5-2 

1-1  +  1 

1+4-3 

1  +  1  +  1 

1_4_|.    5-   2 

-1-1-1 

+  4-16  +  20-8 

+1+1+1 

-   3  +  12-15+   6 

ld=0  +  l±0  +  l 

l±0-14  +  30-23  +  16 

SYNTHETIC  DIVISION. 
(44^5  page  394.) 

-1+2—2+2 


1  —  2  +  2  —  2  +  2,  etc. 
Hence  the  quotient  is  1  —  2^  +  2a;'— 2z'  -|-2ar*— ,  <fec.,  Ans, 


-1+1-1+1 


1  —  1  +  1  —  1  +  1,  &c. 
Hence  the  quotient  is  1  —x  +  x*  —x*  +  x* — ,  <fec.,  Ans, 

3.  1-5  +  10-10  +  5-1  |2-1 

+  2—   6+    6-2 

-  1+    3-2+1 

1  —  3+   3-100 
Henee  the  quotient  is  a' — Sa^x  -\-Sax*— x*,  Ans, 

4.  1-5  +  15-24  +  27-13+5  |2-4  +  2-l 

+  2—   6  +  10 

—  4  +  12-20 

+   2—   6  +  10 
—   1+   3-5 

1-3+   5        0       0       0     0. 
Hence  the  quotient  is  x^—3x-\-5,  Am. 

(391-394) 


CARDAN'S   RULE.  253 


6.  Idb0d=0db0=b0±0±0-1  |1 

1  +  1  +  1  +  1  +  1  +  14-1 


1  +  1  +  1  +  1  +  1  +  1  +  1+0. 
Hence  the  quotient  is     x* + x^y-^-x^y*  +  x^y*  +  x*y*  4-ary' + y*,  Am, 


CARDAN'S  RULE  FOR  CUBIC  EQUATIONS. 
(449,  page  404.) 


4.  We  have  3/)=— 6,     or    7?=— 2; 

2q=z     6,     or     q=z     3; 


fY+F=^9-8=-l. 
Whence  by  formula  (^), 

a;=(3  +  l)*  +  (3-l)^=V4  +  V/2=2.8473+,^«*. 

6.  Here,  3j9=9,     or    ;)=3; 

22'=6,     or     5'  =  3; 


^7+?=  *^^  +  27=f^36=  ±6, 
Whence  by  formula  {A\ 

a;=(3  +  6)' +  (3--6)^=V9 +  ^^=.63783+,  ^n«. 

6.  To  transform  this  equation  into  another  deficient  of  its  second 
term,  according  to  (437),  put  a;=y  — 2,  and  we  shall  have  for  the 
transformed  equation, 

y>_25y  +  66  =  0. 

Hence,  in  applying  Cardan's  rule, 

3;)= -25,     or    p=-%^', 
25-=— 66,     or     5^=— 33; 
f^«+P=V^1089— i-VV^==±22.689737  +  . 
(394-404) 


254  NUMERICAL   EQUATIONS   OF   HIGHER   DEGREES. 

Whence  by  formula  (^), 

y=(  — 33 +  22. 689737)^  + (-33-22. 589737)^ 
=    — 2.18-3.82=-6. 
Therefore,  x=  —  6  —  2=z  —  8. 

Dividing  the  given  equation  by  (a? +  8),  we  have  for  the  depressed 
equation, 

ar'  — 2a;H-3=0; 

whence,  a;=ldbV^— 2. 

Hence,  the  three  roots  are 

—  8,  l+»^-2,  and  l-f^-2,  Ans, 


LIMITS  OF  REAL  HOOTS. 
(453,  page  408.) 

2.  Here,  n=2,  and  /*=25  ;  hence, 

VP  +  l=^^  +  l=6yAn8. 

3.  Supplying  the  deficient  term,  we  have 

x*  +  0x*  —  5x^—9x-\-10  =  0. 
Therefore,  71  =  2,  and  P  —  9;  hence, 

VP  +  l=i^9-hl=iy  Ans. 

4.  Here  n=3,  and  P  =  8;  hence, 

VP  +  l=V8  +  l  =  d,  Ans, 

(454,  page  408). 

1.  Changing  the  signs  of  the  alternate  terms,  we  have 
a-'  +  3z' +  5^—7  =  0. 
Therefore,  n=3,  and  Pr=7;  and 

VP  + 1  =  \/7  + 1  ==  3,  in  whole  numbers,  Ans. 
(404-408) 


HORNER'S  METHOD.  255 

2.  Completing  the  terms,  we  have 

a:*=bOx'  — 15x'— 10i;  +  24  =  0. 
Changing  the  signs  of  the  alternate  terms,  the  equation  becomes 

The  term  having  zero  for  a  coefficient  may  always  be  regarded  as 
positive.     Hence 

w=2,     and  F  =  t5. 

VP  +  l='^^  +  l=5,  Ans. 

3.  With  the  alternate  signs  changed,  the  equation  is 

x'  +  Sx'  +  2x'—2lx'—4x^  +  lz=0. 
Hence,  w=3,     and  P=2l. 

VP  +  1  =  ^^27  +  1  =  4,  ^w«. 


HORNER'S  METHOD   OF  APPROXIMATION. 
(464,  page  420.) 

3.  We  have  given 

A''=a:'  +  2x'  — 23ar— 70. 

The  first  derived  polynomial  is 

X,~3x'-^4x—2S. 
We  multiply  X  by  3,  and  divide  the  result  by  X, .     Thus, 


3a;'  +  6x'— 69j;— 210 
3a;'4-4j;'  — 23r 


3rc'  +  4a:— 23 


X,  +1 


2a;»-46^-210 
a;«_23.r— 105 
3x'— 49-^  — 315,  new  prepared  dividend, 
3a;' +    4.r—    23 

—  73^-292.     Hence,  B=x  +  i, 
(408-420) 


256  NUMERICAL   EQUATIONS  OF  HIGHER  DEGREES. 

We  now  divide  X,  by  B,     Thus, 

3x'  +  12r 


3a:-8 


—  8j:-23 

—  8r— 32 

■f  9.     Hence,  i2,  =  — 9. 

Therefore,  the  functions  are  as  follows : 

X  =z*  +  2f— 23x^10, 
X,=3x*  +  Ax—2S, 
R  =ar  +  4, 
i2,  =  -9. 

Substituting  in  these  functions,  x=  —  qo  and  x=z  +  oo  successively, 
we  have  the  following  results,  in  respect  to  signs : 


For 


(  j:=  +  oo,  -f- 


~      +      —      — »     2  variations. 
+      — ,     1  variation. 


Hence  the  given  equation  has  1  real  root. 

If  we  substitute  x=0  in  the  functions,  the  signs  will  be 

-      -      +      -, 

giving  2  variations.  Hence  the  real  root  must  lie  between  0  and 
+  oo ;  or,  it  is  positive.  To  ascertain  its  position,  make  a;=l,  aj=2, 
ar=3,  etc.,  successively. 


For' 


Hence,  the  initial  figure  is  5.    The  decimal  part  is  found  by  the 
following  operation : 


x=\,  signs. 

— 

— 

+ 

— , 

2  variations. 

ar=2,     " 

— 

— 

+ 

— , 

2 

(( 

X=:3,      « 

— 

+ 

+ 

— , 

2 

(( 

ar=4,     « 

— 

+ 

+ 

— , 

2 

i( 

x=b,     " 

— 

+ 

+ 

— , 

2 

u 

ir=6,     « 

+ 

+ 

+ 

— , 

1  variation. 

(420,  Ex.3) 


HORNER'S   METHOD. 


257 


+  2 

_5 

7 
5 

12 
5 

(^)  17.0 
.1 

17.1 
.1 

17.2 
.1 


(2)  17.30 
3 

17.33 
3 

17.36 
3 


(3>  17.390 
4 


17.394 
4 

17.398 
4 

<*>  17.402 
<»>  17.4 

{«)  1 


—  23 
35 
12 
60 

0>  72.00 
1.71 


73.71 

(2) 

75.4300 
.5199 

75.9499 
.5208 

(8) 

76.470700 
69576 

72.540276 
69592 

(4) 

76.609868 
8701 

76.618569 
8701 

(5) 

76.62727 
122 

76.62849 
122 

(d) 

76.6297 

1 

76.6298 
1 

(7>  76.630 

<^>  76.63 

(9>  76.6 

(10)  77 

(420,  Ex.  3 ) 


-70  |5.1345787253 
60 


<^>— 10.000 
7.371 


C2)  — 2.629000 
2.278497 


(8)  — .350503000 
.306161104 

(*>— 44341896 
38309285 


(5)  _  603  26 11 
5363994 

(«>  — 668617 
613038 


(7)-55579 
53641 


(8)  — 1938 
1533 


(9)_405 
383 


(10)_22 
23 


'Z58  NUMERICAL   EQUATIONS  OF   HIGHER   DEGREES. 

4.  AVe  have  given 

X=ar'— x'  +  70x-300. 
The  first  derived  polynomial  is 

Multiplying  X  by  3   to   avoid  fractions  in  dividing,  we  proceed  as 
follows : 

3j:'  — 3x'  +  210ar—   900\3z^  —  2x-\-l0 


32r'  — 2:r'+    70a; 


k  -1 


—  a;'  +  140i:—   900 

--3z'  +  420-c  — 2700,  prepared  dividend. 

—  3a:'+-      2x~      70 

+  418X— 2630.     Hence,  i2=:—209j:  + 1315. 
To  avoid  fractions  in  the  next  division,  multiply  X,  by  209. 


627a:'-   418^+      14630 
627ar'  — 3945.C 


209a:  +  1315 


-3r, 


3527 


-f3527.r+      14630 

Multiplying  by  209,      +737 143a:— 3057670 

+  737143j:-4038005 


+  1580335. 

i2,  =  -1580 

ence,  the  functions  are 

X  =     a;' -a:' +  70a:- 300 

1 

X,=     3a;' -2a: +  70, 

H  =-209a:  +  1315. 

i2,  = -1580335. 
Let  x=—  cc\  we  have    —      +      +     —  , 

2 

variations ; 

«    a:=  +  00,       "             +      V      ^      _  , 

1 

variation  ; 

"    ^=     0,          "             _      +      +      _  , 

2 

variations ; 

Hence,  the  given  equation  has  but  one  real  root,  and  this  lies  be- 
tween 0  and  +  oo  ;  it  is  therefore  positive. 

Leta:=l;  we  have     —      +      +      —  , 

«    x=2  "  _      +      4-      —  , 

«    x=3  "  —      +      +      _  , 

"    a:=4  "  4-      +      +      _  , 

(420,  Ex.  4  ) 


variations, 


HORNER'S  METHOD. 


259 


Hence,  the  initial  figure  is  3.     The  decimal  part  is  found  by  the 
following  operation : 


1  —1 

f  70 

-300  |3.7387936878 

3 

0 

(2) 
(8) 

(4) 
(5) 
(6) 

6 

76 
15 

228 

2 
3 

0)- 72.000 
67.963 

5 
3 

)  91.00 
6.09 

(2)  — 4.037000 
3.119217 

0)  8.0 

.7 

97.09 
6.58 

(8)  — .917783000 

.834882272 

8.7 

.7 

103.6700 
.3039 

(4)  _  82900728 
73114357 

9.4 

7 

103.9739 
3048 

(5)  — 9786371 
9401144 

(2)  10.10 
3 

104.278700 
81584 

(6)-385227 
313374 

10.13 
3 

104.360284 
81648 

(7)-71853 
62675 

10.16 
3 

104.441932 
7150 

(^)-9l78 
8357 

(S)  10.190 

8 

104.449082 
7150 

(9) -831 
732 

10.198 
8 

104.45623 
92 

ao)-89 
84 

10.206 
8 

104.45715 
92 

—5 

W  10.214 

104.4581 

(5)  10.2 

(7) 

104.458 

C«)  1 

(8) 

104.46 

(9) 

104.5 

(10) 

105 

(420,  Ex.4) 

260 


NUMERICAL    EQUATIONS   OF    HIGHER   DEGREE3. 


5.  We  have  given 

X=a:'+a;'-500. 
The  first  derived  polynomial  is 

Multiply  X  by  3  to  avoid  fractions  in  dividing. 


3a;' +  3a;"-- 1500 
Sx"4-2a; 


Sx'-{-2x 


X,  +1 


a;'  — 1500 
3jr'-4500 
3a;'  +  2a; 


3ar'+        2a; 
3ar'  + 6750a; 


—  2a;  -4500 
a; +  2250 


Hence,  i2=a;  + 2250 


3a;— 6748 


-6748a; 
-6748jr- 15183000 


+  15183000.     Hence,  i2,  =  —  15183000. 


Thus  the  functions  arc 


X  = 

R  = 

Let  x=.  —  oo ;  we  have 

**  ar=:  +  O) ;   " 


0: 


a:'+a;'— 500, 
3x'  +  2^-, 
a; +  2250, 
-15183000. 

-  +   - 

+   +   + 

-  ±   + 


2  variations. 

1  variation. 

2  variations. 


Hence,  there  is  but  one  real  root ;  and  this  is  positive,  because  it 
lies  between  0  and  +  QO. 


.et 

a;=l; 

signs. 

— 

+ 

+ 

— , 

2 

variations. 

11 

arz:r2; 

<( 

— 

+ 

+ 

— , 

2 

a 

u 

a;=3; 

It 

— 

+ 

+ 

— , 

2 

(( 

u 

a;=:4 

u 

— 

+ 

+ 

— , 

2 

u 

u 

a;=5 ; 

u 

— 

+ 

+ 

— , 

2 

(( 

u 

a;=6; 

(( 

— 

+ 

+ 

— , 

2 

u 

t( 

a;=7; 

i( 

— 

+ 

+ 

— , 

2 

u 

u 

a;=8; 

u 

+ 

+ 

+ 

— , 

1 

variation. 

(420,  Ex.  5) 


HORNER'S  METHOD. 


261 


Hence,  the  initial  figure  is  7  ;    the  decimal  part  is  found  as  fol- 
lows : 


+  1 

0 

-500  17.6172797559 

1 

56 

392 

8 

66 

W— 108.000 

7 

105 

104.736 

15 

(1) 

161.00 

(2)  — 3.264000 

7 

13.56 

1.887181 

0)  22.0 

174.56 

(8)-1.376819000 

.6 

13.92 

1.323862113 

22.6 

(2) 

188.4800 

W  — .052956887 

.6 

.2381 

37858967 

23.2 

188.7181 

(5)- 15097920 

.6 

.2382 

13251090 

(2)  23.80 

(8) 

188.956300 

(6)_i846830 

.01 

.166859 

1703728 

23.81 

189.123159 

W- 143102 

.01 

.166908 

132512 

23.82 

(4) 

189.290067 

(S)-10590 

.01 

4770 

9465 

(8)  23.830 

189.294837 

(9)-1125 

.007 

4770 

947 

23.837 

(5) 

189.29961 

(io)_n8 

.007 

167 

170 

23.844 

189.30128 

-8 

.007 

167 

W  23.851 

(«) 

189.3029 
2 

(5)  23.9 

189.3031 
2 
189.303 

(«)  2 

C) 

(8) 

189.30 

(9) 

189.3 

(10) 

189 
(420,  Ex.  6) 

262  NUMERICAL   EQUATIONS  OF   HIGHER  DEGREES. 

6.  We  have  given, 

X=ar'— a:'  — 40ar— 108. 
The  first  derived  polynomial  is 

X,=dx'-2x-40, 
Multiply  Xby  3,  to  avoid  fractions  in  dividing. 

3a;'  — 32:'— 120j:4-324j3j:'  — 2ar— 40 
3jr'  — 2ar'—   40.c  Lr,  —1 


Multiply  by  3, 


—  242.1:4-932.      Hence,  i2=121ar— 46«, 
Multiplying  X,  by  121,  to  avoid  fractions  in  the  next  division, 


-   a:'- 

80a:  4-324 

-3a:'- 

240a:  4- 9V2 

-3a:'  + 

2a:  4-    40 

363a:'—      242a:r-      4840 
366.r'-   1398a: 

+  1156a:-  4840 
Dividing  by  4,  4-      289.i:—      1210 

Multiplying  by  121,  4- 34969a;— 146410 

4- 34969a:—  134674 


121a:— 466 


3a:,  4-289 


11736  =— J?. 


Thus  we  find 


X  =a:'-a:*— 40a:4-108, 

X,=3a:'— 2a:-40, 

jR  =121a:— 466, 

i2,  =  + 11736. 
Let  a:=  —  oo ;  we  have     -^     4-      —      +,3  variations 
"    a:=4-  oo;        "  +      +      +      +,     o 

"    ar=       0;        «  +      _      _      +^     2         « 

Hence,  there   are  three  real  roots,  two  between  0  and  4-  co,  and 

and  one  between  0  and  —  oo  ;  that  is,  there  arc  two  positive  roots, 
and  one  negative  root. 

To  find  the  situation  of  the  positive  roots,  let 

a:=r:l  ;  signs,  4-  —      —      +»     2  variations. 

x=2;      "  +  _      _      4-,     2 

x=d;      «  +  _     _      4-,     2 

a:=4;      "  —  ±      +      +,     1   variation. 

x=5;      "  4-  +      +      4-,     0 

Hence  the  initial  figures  of  the  positive  roots  are  3  and  4. 
To  find  the  situation  of  the  negative  root,  let 
(420,  Ex.  6) 


HORNER'S  METHOD. 


263 


ar=-2;  signs, 

+ 

— 

— 

-f, 

2  variations. 

ar=-4;      " 

4- 

4- 

— 

+  , 

2            " 

x=-e;      " 

+ 

+ 

— 

+  1 

2            " 

ar=-8;      « 

— 

+ 

— 

+, 

3            " 

Hence  the  negative  root  is  situated  between  6  and  8. 

Let  a:  =—6;  signs,  +      +      —      +>     2  variations. 

"    ar=-7;      "  _      4-      _      +,     3 

Hence,  the  initial  figure  of  the  negative  root  is  —  6, 
The  decimal  part  of  the  first  root  is  obtained  thus  : 


-1 

-40 

+  108  |3.3792053825 

3 

6 
—  34 

-102 

2 

0)          6.000 

3 

15 

-4.953 

6 

(1)- 19.00 

(2)     1.047000 

3 

2.49 
-16.51 

-.931147 

0)  8.0 

(8)     .115853000 

.3 

2.58 

-.113285061 

8.3 

(2)  — 13.9300 

(4)     2567939 

.3 

.6279 

—  2500650 

8.6 

-13.3021 

(5)       67289 

8 

<2)  8.90 

.6328 
(8)-12.669300 

-62507 
(6)       4782 

7 
8.9V 

82071 
-12.587229 

-3750 
(T)     1032 

7 

82152 

-1000 

9.04 

C4)_12.505077 

(8)     32 

1 

1827 

-25 

(8)  9.110 

-12.503250 

(9)      7 

9 

1827 

-7    • 

9.119 
9 

(5)- 12.50142 
5 

0 

9.128 
9 

-12.50137 
5 

W  9.137 

(6)- 12.501 

(5)  9.1 

(7)- 12.50 
<S)-12.5 

(9) -13. 

(420,  Ex. 

6) 

264 


NUMERICAL    EQUATIONS   OF   HIGHER    DEGREES. 


The  decimal  part  of  the  second  root  is  obtained  thus : 


—1 

-40 

+  108|4.5875359541 

4 

12 

-112 

3 

-28 

(1) -4.000 

4 

28 

2.875 

1 

0)  0.00 

W- 1.125000 

4 

5.75 

1.020512 

(1)  11.0 

6.75 

(3)— .104488000 

.5 

6.00 

97009003 

11.5 

(2)  11.7500 

(4)_  7478997 

.5 

1.0064 

6977044 

12.0 

12.7564 

(5)-501953 

.5 

1.0128 

418826 

C2)  12.50 

(8)  13.769200 

(6)-83127 

8 

89229 

69807 

12.58 

13.858429 

(7)- 13320 

8 

89278 

12565 

12.66 

W  13.947707 

(3) -755 

8 

6381 

698 

(8)  12.740 

13.954088 

W-57 

7 

6381 

56 

12.747 

(5)  13.96047 

(10)  _i 

7 

38 

1 

12.754 

13.96085 

0 

7 

38 

W  12.761 

(«)  13.9612 

1 

13.9613 

(»)  12.8 

W  13.96 

1 

(»)  14.0 

(«)  1 

(7)  13.961 

(10)  14. 

Note. — To  obtain  the  trial  figure  for  the  second  place  in  the  root,  proceed 
according  to  Note  5,  (464).     Thus,  we  shall  have 


=/-f=/^:=^-- 


By  trial,  we  find  that  this  must  be  diminished  by  .1,  making  .6  for  the  second 
figure  of  the  root. 

(420,  Ex.  6) 


Horner's  method. 


265 


To  obtain  tlie  negative  root,  we  may  change  the  signs  of  the 
alternate  terms ;  the  result  obtained  will  then  be  positive,  according 
to  Note  3,  (464:).  This  is  not  necessary,  however;  for  we  niay 
derive  the  negative  root  directly  fiom  the  given  coefficients,  as  will 
be  seen  from  the  following  operation  : 


W-21.880 

—  6 

—  21.886 
— C 

—  21.892 

—  6 

")-2].898 

^5)^21.9 

(6)_2 


-40 

+  108|— 6.9667413367 

+  42 

-12 

2 

0)  96.000 

IS 

-88.119 

0)  80.00 

17.91 

97.91 

18.72 

(2) 

116.6300 

1.3056 

117.9356 

1.3092 

(8) 

119.244800 

.131316 

119.376116 

.131352 

(4) 

119.507468 

15329 

119.522797 

15329 

(5) 

119.53812 

87 

119.53899 

87 

(«)  119.5399 
(7)  119.540 
W  119.54 
<9)  119.5 
(10)  120 


—  7.076136 

(8)     .804864000 

—.716256696 

W     88607304 

—  83665958 

i^)     4941346 

—  4781560 

(6)     159786 


4384 
3586 


(9) 


798 
-717 

OO)     81 

—  83 

8 


(420,  Ek.  6) 


266  NUMERICAL   EQUATIONS   OF   HIGHER   DEGREES 

7.  We  have  given 

X=a:'-4x'  — 24j:  +  48. 
The  first  derived  polynomial  is 

X,=z3x'-8x-2i. 
Multiplying  X  by  3  to  avoid  fractions  in  dividing, 

3.r'  — 8a;— 24 


3j:'-12a:'-72.r+144 
3.c'—   8x'  — 24.r 

—  4j:''  — 48j;+144 
Multiplying  by  3,                   __   s^'  — 36^+108 

—  3r'+    8x-\-    24 

-442-+    84.  llence,ijj=lla:— 21. 
Multiplying  X,  by  11,  to  avoid  fractions, 


33j;'—   882:—   264 

llar-21 

33j:'—    Q3x 

Sx,  -25 

—   25x—   264 

Multiplying  by  11,                —  27oj;— 2904 

—  215X+   525 

-3429.     Hence,  JR,  =342 

Hence,  we  have  the  functions, 

X  =x'-4jr'- 242^  +  48, 

X,  =3j:'-8a:-24, 

E   =ll2r-21, 

i2,=3429. 

Leta:=-Qo;  we  have      —      4-      —      +,3     variations 

"    a:=+  00;       "              4-      4-      +      4-,     0 

"   ar=        0;       "              4-      _      _ 

+  ,     2 

Hence,  there  are  three  real  roots,  —  two  situated  between  0  and 
+  00  and  therefore  positive,  and  one  situated  between  0  and  —  00, 
and  therefore  negative. 


hetx= 

1; 

we  have 

+ 

— 

— 

+  , 

2 

variations. 

«    x= 

2; 

(( 

— 

— 

+ 

+  , 

(( 

"   x= 

3; 

(( 

— 

— 

+ 

+  , 

u 

"    x= 

4; 

(i 

— 

— 

+ 

+  , 

u 

«    x= 

5; 

u 

— 

+ 

+ 

+  , 

u 

«   x= 

6; 

u 

— 

+ 

+ 

+  , 

u 

"    x= 

7; 

u 

+ 

+ 

+ 

+  , 

0 

u 

«   x=- 

-1; 

u 

+ 

(420 

,  J^x 

•7) 

+  , 

2 

u 

Horner's  method. 


267 


Let  ar=~2;  we  have 

+ 

+ 

— 

+  , 

2 

variations. 

"    ar=  — 3;         " 

+ 

+ 

— 

+  , 

2 

(t 

"    a:=  — 4;          " 

+ 

+ 

— 

+  , 

2 

(( 

"    x=-5\ 

— 

+ 

— 

+  , 

3 

(( 

By  inspecting  the  column  of  variations,  we  perceive  that  one  root 
is  between  1  and  2,  another  between  6  and  7,  and  another  between 
—  4  and  —5.     Hence,  the  initial  figures  of  the  three  roots  are 

1,     6,     -4. 

The  first  root  is  obtained  as  follows  : 


—  4 

-24 

+  48  |1.7191292611 

1 
—  3 

-   3 

-27 

(l)~ 

27 
21.000 

1 

-  2 

— 

20.447 

-2 

0)-29.00 

(2) 

.553000 

1 

-     .21 
-29.21 

(8; 

—  .289189 

a)~i.o 

>     .263811000 

.1 

+      .28 

-.260077041 

—  .3 

.1 

(2) -28.9300 
+          111 

W     3733969 

—  2888700 

+  A 

,1 

—  28.9189 
-f         112 

(5)     845259 

-577737 

(2)    1.10 

(3)-28.907700 

(6)     267522 

1 

4-         10251 

-259981 

1.11 

—  28.897449 

(7)     7541 

1 

+         10332 

-5777       ' 

1.12 

W-28.887117 

(8)     1764 

1 

+             116 

-1733 

(8)  1.130 

-28.887001 

(9)     31 

9 

+             116 

-29 

1.139 

(S)-28.88688 

(10)     2 

9 

1.148 

9 

+                2 
—  28.88686 
4-                2 

(S) 

•     -3 

-28.89 

<4)  1.157 

(6)-28.8868 

(9) 

-28.9 

(5)  1.2 

(7) -28.887 

(420,  Ex.7) 

(10) 

-29. 

268  NUMERICAL   EQUATIONS  OF   HIGHER   DEGREES. 


The  decimal  part  of  the  second  root  is  found  thus  : 

+  48  |6.546145'7261 
-72 


1 

-4 

6 

■1  « 

2 

6 

8 

6 

(1) 

14.0 

.5 

14.6 

.5 

15.0 

.5 

(2) 

15.50 

4 

15.54 

4 

15.58 

4 

(8) 

15.620 

6 

15.626 

6 

15.632 

6 

(4) 

15.638 

(5) 

15.6 

(6) 

2 

-24 

12 
48 


Oy  36.00 
7.25 


43.25 
7.50 


<«>  60.7500 
.6216 


51.3716 
.6232 


<3>  51.994800 
93756 

52.088556 
93792 

<*>  52.182348 
1564 

52.183912 
1564 


(5)  52.18548 

62 

52.18610 
62 

(6>  52.1867 
1 


52.1868 
1 


(^  52.187 
(5>  52.19 
<9>  52.2 
(10)  52 

(420,  Ex. 


(i>-24.000 
21.625 


C2)  — 2.375000 
2.054864 


(8> -.3201 36000 
.312531336 


(4)  _  7604664 

5218391 

(5)  — 2386273 

2087444 

c«>- 298829 
260934 


(7) -37895 
36531 


(8) -1364 
1044 


t9)  — 320 
313 

(10)  _  7 

5 

—  2 


n 


HORNERS  METHOD. 


269 


.     And  the  decimal  part  of  the  third  root  is  found,  without  chang- 
ing the  signs  of  the  alternate  terms  of  the  equation,  as  follows : 


1     -4 

-24 

+  48   -4.2652749871 

-4 

+  32 

-32 

-8 

8 

0)     16.000 

-4 

48 
0)  56.00 

—  11.848 

—  12 

(2)     4.152000 

—   4 

3.24 

—  3.811176 

(l)-16.0 

59.24 

<3)     .340824000 

—      .2 

3.28 

—  .323033625 

-16.2 

(2)  62.5200 

W     17790375 

—     .2 

9996 

-12938807 

-16.4 

63.5196 

(5)     4851568 

—     .2 

1.0032 

-4528900 

(2)- 16.60 

(3)  64.522800 

(6)     322668 

-        6 

83925 
64.606725 

-258799 

-16.66 

0)     63869 

-         6 

83950 

—58230 

—  16.72 

W  64.690675 

(8)     5639 

-         6 

3359 
64.694034 

-5176 

W- 16.780 

(9)     463 

-           5 

3359 

-453 

-16.785 

(5)  64.69739 

(10)     10 

—           5 

118 

-1 

—  16.790 

64.69857 

3 

—           5 

118 

(*)- 16.795 

(6)  64.6997 
1 

(»J-16.8 

64.6998 
1 
(7)  64.700 

(«)— 2 

(8)  64.70 

(9)  64.7 

00)  65. 

(420,  Ex.7) 

*270  NUMERICAL   EQUATIONS   OF    HIGHER   DEGREES. 

4a:*  +  4j:'4-4j;'—    4^—2000    4a:' +  3;c' +  2ar— 1 


8,     Given, 
Whence, 


X,  +1 


or, 


a:'  +  2a:'-  82:— 2000 
4a;' +  8j:'  — 12a; -8000 
4x'  +  3a;"+    2a;—       1 

5a;''-14a;-7999.        i?=-5a;«  + 14a; +  7999. 


or. 


20a;' +    15a:' +  10a:— 

20a:'—   56a;'—   31996a; 

71a;' +   32006a-— 
855a;'  + 160030a; — 


—  5a;'  +  14a;-f-7999 


-4a;,  -71 


5 
25 


355a;'—       994a;— 567929 


161024a;  +  567904 


Whence, 

Multiplying  R  by  5032, 

—  25160a;' +    70448a-  + 

—  25160a;'—   88735a; 


i2.=—5032j:— 17747, 


40250968 


-5032a;-l7747 


5a;,  —159183 


or. 


159183a;+  40250968 

5032  •159183a;  +  202542870976 
6032 -1591833;+      2825020701 


+  199717850275  =— i2j 


In  these  functions,  let 

a;=  —  00 ;  we  have 
a;=  +  00 ;        " 
x=        0;        « 


+      —      —      +      — ,     3  variations. 
+      +      —      —      — ,     1  variation. 
—      —      +      —      — ,     2  variations. 

Hence,  the  given  equation  has  one  real  root  between  0  and  00, 
which  must  be  positive ;  and  one  real  root  between  0  and  —  00, 
which  must  be  negative.  By  proper  substitutions,  we  shall  find  the 
initial  figures  to  be  4  and  —4. 

(420,  Ex.  8) 


HORNER'S   METHOD.  271 

The  decimal  part  of  the  positive  root  is  found  as  follows : 


+1 

4 

+  1 
20 

21 
36 

57 
52 

0)  109.00 
6.96 

—  1 

84 

83 

228 

—  500  14.4604168201 
332 

5 

4 

(1)  — 168.0000 
142.9536 

9 
4 

(1)311.000 
46.384 

(2)— 25.04640000 
24.87028656 

13 
4 

357.384 
49.232 

(8)-. 17611344 
.16900578 

(1)17.0 
.4 

115.96 
7.12 

(2)406.616000 
7.888776 

(*)- 710766 
422569 

17.4 
.4 

123.08 

7.28 

414.504776 
7.956168 

(5)  — 288197 
253543 

17.8 
.4 

(2)130.3600 
1.1196 

(•^)  422.460944 
53495 

(6)— 34654 
33806 

18.2 
.4 

131.4796 
1.1232 

132.6028 
1.1268 

422.514439 
53495 

(■)-848 
845 

(2)18.60 
.06 

('t)  422.5679 
13 

(8) -3 

4 

18.66 
.06 

(8)133.7296 
75 

422.5692 
13 

18.72 
.06 

133.7271 
75 

(•"')  422.571 
(6)422.57 

18.78 
.06 

133.7446 
75 

(7)  422.6 

(8)  423 

(8)18.84         (4)134. 

Note. — The  first  contracted  terms  in  the  operation,  marked  (4),  occur  in  con- 
nection with  the  cipher  in  the  root ;  and  the  pupil  will  observe  that  they  are 
therefore  contracted  twice  the  usual  number  of  places. 

(420,  Ex.  8) 


272 


NUMERICAL    EQUATIONS   OF   HIGHER    DEGREES. 


To  obtain  the  decimal  part  of  the  negative  root,  change  the  signs 
of  the  alternate  terms,  and  proceed  as  in  the  following  operation.: 


-1 

+  1 

+  1 

-500  14.9296646474 

4 

12 
13 

52 
63 

212 

3 

(l)-288.0000 

4 

28 

164 

275.7411 

7 

41 

0)  217.000 

(2)-12.25890000 

4 

44 

89.379 

8.23961296 

11 

0)  85.00 

306.379 

(3) -4.0 1928704 

4 

14.31 

102.987 

3.74208814 

0)  15.0 

99.31 

(2)  409.366000 

(4)-.27919890 

.9 

15.12 

2.614648 

.25023185 

15.9 

114.43 

411.980648 

(5)-2696705 

.9 

15.93 

2.622104 

2502841 

16.8 

(«)  130.3600 

(3)  414.602752 

(6) -193864 

.9 

.3724 

1.184819 

166859 

17.Y 

130.7324 

415.787571 

(^)- 27005 

.9 

.3728 

1.186331 

25029 

(2)18.60 

131.1052 

W  416.97390 

(8) -1976 

2 

.3732 

7919 

1669 

18.62 

(^)   131.4784 

417.05309 

(^-307 

2 

.1681 

7920 

292 

18.64 

131.6465 

(5)  417.1323 

(io)_iv 

2 

1681 

79 

17 

18.66 

131.8146 

417.1402 

0 

2 

.1681 

79 

CS)18.68 

(4)  131.98 

(<5)  417.148 

W2 

1 

(')  417.15 

131.99 

(3)  417.2 

1 

(9)  417.  Hence, 

,  -4.9296646474,  root. 

132.00 
1 
(5)  132. 

(10)  42 

(6)1 

(420,  Ex.  8) 


<^^'^^ 


HORNER  S  MET] 


9.  We  have  given 


TY, 


k..-9x.-n..-2o>^££fOR^ 


The  first  derived  polynomial  is 
Multiplying  X  by  4,  to  avoid  fractions  in  dividing, 


4a;*-362:'-   44ar'- 
4x*-2lx'—   22^'- 


8Qx-\-  16 
202r 


4^'  — 27a?'— 22jr— 20 


9 


—  f)x'—   22r'—   60x+    16 
or      -36x'—   88a;'  — 240jr+    64 

—  36a:'  +  243a:'  +  198a:  +  180 


-331a;'-438a:- 

-116. 

tiplying  X,  by  331, 

1324a:'—   893Ya:'- 

7282a;- 

1324a:' -f-    1752a:' + 

464a-- 

i2=331a:'  +  438a;  +  116. 


—  10689a:'- 


6620 


7746a;—        6620 


331a:' +  438a: +  116 


4a:,— 10689 


or         — 331*  10689t'— 2563926a;— 2191220,      prepared  dividend, 
—  331-10689.r'— 468l782a:-1239924 


+  21l7866a:-    951296 

Dividing  by  32,  and  changing  signs, 

i?,  =  — 66183ar  +  29728. 
Multiplying  i2by  66183, 


66183- 331a:' +  28988154a;4- 
66183-331a:'—   9839968a;  + 


7677228 


-66183.r  4-29728 
-331a-,-19414061 


+  38828122a:+  7677228 

or,  +19414061a;+  3838614 

or,  +66183 -19414061a: +  254050990362,  prepared  dividend, 

+  66183 -1941406^—577141205408 


+  831192195770=-/?,. 
(420,  Ex.  9) 


274  NUMERICAL   EQUATIONS   OF  HIGHER   DEGREES. 

Thus  we  have,  for  the  several  functions, 

X  =     x*—9x*—Ux^  —  20x-\-4, 
X,  =     4x'  —  2lx'-22x-20, 
R  =     331a:'  +  438a;+116, 
i2,=— 66183ar  +  29728, 
7?3  =  -831192195770. 

To  find  the  number  of  real  roots,  let 


x=—  OO;  signs, 

+ 

N 

+ 

+ 

— ,     3  variations. 

ar=4-  Oo;      " 

+ 

+ 

4- 

— 

— ,     1  variation. 

x=        0 ;      " 

+ 

— 

+ 

+ 

— ,     3  variations. 

Hence,  there  are  two  real  roets  between  0  and  oo.     To  find  their 
situation,  let 


x=  0 

signs. 

+ 

— 

+ 

+ 

— 

,     3  variations. 

Xz=    1 

u 

— 

— 

+ 

— 

— 

,     2 

u 

x=   2 

t( 

— 

— 

+ 

— 

— 

,     2 

u 

x=  3 

(( 

— 

— 

+ 

— 

— 

►     2 

u 

«=  4 

ti 

— 

— 

+ 

— 

— 

2 

a 

ar=  5; 

u 

— 

— 

+ 

— 

— , 

2 

(i 

x=  6 

u 

— 

— 

+ 

— 

— 

.     2 

a 

x=  7 

t( 

— 

— 

+ 

— 

— 

2 

(( 

x=  8; 

u 

— 

+ 

+ 

— 

— 

2 

(( 

«=   9 

u 

— 

+ 

+ 

— 

— 

2 

(( 

ar=10; 

u 

— 

4- 

+ 

— 

— , 

2 

(( 

ar=ll 

u 

+ 

+ 

+ 

— 

— , 

1  variation. 

Hence  the  lesser  root  is  situated  between  0  and  1,  and  the  greater 
between  10  and  11. 

The  initial  figures  of  the  greater  are  10.     To  find  the  initial  figure 
of  the  lesser,  let 


x=,l ;  signs, 
x=.2:      « 


+      - 


3  variations. 

2  " 


Hence,  the  initial  figure  is  .1.     The  decimal  part  of  this  root  is 
as  follows : 

(420,  Ex.  9) 


HORNER'S  METHOD. 


275 


1   —9.0 

-11. 

-20. 

+  4  1. 17968402504 

.1 

—  .89 

—  11.89 

-  1.189 

—  21.189               ( 

-2.1189 

—  8.9 

■^)     1.88110000 

.1 

-      .88 
-12.77 

-   1.277 

—  1.64238179 

—  8.8 

(1) -22.466000 

(2)     .238718210000 

.1 

-     .87 

—     .996597 
-23.462597 

-.221760635319 

-8.7 

0)_  13.6400 

(8)     16957574681 

.1 

-     .5971 
-14.2371 

—   1.038051 

—  14873731847 

a)-8.60 

(2)-24.500648000 

W     2083842834 

7 

—  .5922 

—  14.8293 

—      .139422591 

-1984015679 

—  8.53 

-24.640070591 

(5)     99827155 

7 

—     .5873 

-     .140095053 

-99206044 

—  8.46 

(2)-15.416600 

(3)-24.780165644 

(6)     621111 

7 

—     .  74799 

-           9387434 

-496031 

—  8.39 

—  16.491399 

—  24.789553078 

(^)     125080 

7 

-     .  74718 

-           9390416 

-124008 

(2)_8.320 

—  15.566117 

W— 24.79894349 

(8)     1072 

9 

-     .  74637 

-           125250 

-992 

-8.311 

(8)-15.640754 

-24.80019599 

80 

9 

-     .     4970 

-           125255 

—  8.302 

-15.645724 

(5)-24.8014485 

9 

-  .     4970 

—  15.650694 

-               626 

—8.293 

-24.8016111 

9 

-     .     4970 

-                626 

(8)  — 8.284 

W- 15.6557 

(6)-24.801574 

<*>— 8. 

—            6 

(7)-24.8016 

—  15.6563 

—  6 

(8) -24.802 

-15.6569 

-             6 

(5)-15.66 

Note — The  term  marked  (6),  in  the  second  column  from  the  right,  is  contracted  two 
places,  to  obtain  the  next  term  (7),  This  is  on  account  of  the  cipher  which  occurs  in 
the  corresponding  part  of  the  root. 

(420,  Ex.  9) 


276 


NUMERICAL    EQUATIONS   OF    HIGHER   DEGREES. 


The  decimal  part  of  the  greater  root  is  found  as  follows 


-9 

-11 

—  20 

+  4  |10.2586086356 

10 

10 

-1 
110 

109 
210 

-10 

—  30       ' 
1090 

—  300 

1 
10 

0)-296.0000 
225.0096 

11 
10 

0)  1060.000 
65.048 

(2)- 70.99040000 
60.41618125 

21 
10 

0)319.00 
6.24 

1125.048 
66.304 

(«)- 10.57421875 
9.82494438 

0)31.0 
,2 

325.24 
6.28 

(2)1191.352000 
16.971625 

(4)-.74927437 
.73864151 

81.2 
.2 

331.52 
6.32 

1208.323625 
17.051375 

(5) -1063286 
985022 

31.4 
.2 

(2)337.8400 
1.5925 

(5)1225.375000 
2.743048 

(6)-78264 
73877 

31.6 
.2 

339.4325 
1.5950 

1228.118048 
2.745096 

(^-4387 
3694 

W  31.80 
5 

341.0275 
1.5975 

(*)  1230.86314 
.20605 

(8) -693 
616 

31.85 
5 

(8)342.6250 
.2560 

1231.06919 
.20606 

(»)-77 
74 

31.90 
6 

342.8810 
.2560 

(5)1231.2753 
27 

-3 

31.95 
5 

343.1370 
.2560 

1231.2780 

27 

(3)  32.00 

(4)343.39 

2 

343.41 

2 

343.43 

2 

(5)  343 

(6)1231.28 

W3 

(7)1231.3 

(8)1231 

(»)123 
(420,  Kx.  9) 

HORNERS  METHOD. 


277 


10.  We  readily  find  the  functions  to  be 

X  =:c*-12x'  +  12j-3, 

R  =2i;'-3jr+l, 


3  variations. 
0         " 


Let  ar=— oo;  we  have     4-     —     +     —      +, 
"    «=+  oo;       "  +      +      +      4-      +, 

Hence  the  roots  are  all  real.  And  since  the  signs  of  the  given 
equation  give  three  variations  and  one  permanance,  three  of  the 
roots  are  positive,  and  one  is  negative,  ( 447  ). 

To  find  the  situation  of  the  roots, 


Let  xz=z     0 ; 

we  have 

— 

+ 

+ 

— 

+  , 

3  variations. 

**   «=     1; 

(4 

— 

— 

± 

+ 

+  , 

1  variation. 

«   «=     2; 

(( 

— 

— 

+ 

+ 

+  , 

1 

"   x=     3; 

t( 

+ 

+ 

+ 

+ 

+  , 

0         " 

"   ar=~l; 

(i 

— 

+ 

+ 

— 

+  , 

3  variations. 

«   ar=-2; 

i( 

— 

+ 

+ 

— 

+  , 

3 

"    ar=— 3; 

(( 

— 

— 

+ 

— 

+  , 

3         " 

"    ar=-4; 

(( 

+ 

— 

+ 

— 

+  , 

4         " 

Hence,  there  are  two  roots  situated  between  0  and  1,  one  between 
2  and  3,  and  one  between  —3  and  —4. 

To  find  more  definitely  the  situation  of  the  roots  between  0  and  1, 


Let  x=.l  ; 

we  have 

— 

+ 

+ 

— 

+  , 

3  variations. 

"    x=.2; 

a 

— 

+ 

+ 

— 

+  , 

3 

(( 

"    x=.Z\ 

u 

— 

4- 

+ 

— 

+  , 

3 

t( 

"    x=.4; 

a 

— 

+ 

+ 

— 

+  , 

3 

u 

«    «=.5; 

u 

+ 

+ 

± 

— 

+  , 

2 

u 

"    ^=.6; 

u 

+ 

— 

— 

+ 

+  , 

2 

i( 

«    x=.T; 

u 

— 

— 

— 

+ 

+  , 

1  variation. 

Hence,  there  is  one  root  between  .4  and  .5,  and  one  between  .6 
and  .7. 

The  initial  figures  of  the  four  roots,  taken  in  the  order  of  their 
values,  are  as  follows  : 

2,     .6,     .4,     -3. 
(420.  Ex.  10) 


278  NUMERICAL   EQUATIONS  OF   HIGHER   DEGREES. 

The  decimal  part  of  the  first  root  is  found  as  follows  : 


1+0 
2 

—  12 

4 

—  8 

8 

0 
12 

+  12 
-16 

-  4 

0 

—  8  12.858083082 
—8 

2 

2 

0)-11.0000 
8.9856 

4 

2 

15.232 

(2)  _  2.0 1440000 
1.71940625 

6 
2 

0)  + 12.00 
7.04 

19.04 
7.68 

+  11.232 
21.376 

(3)-.29499375 
.29192888 

0)  8.0 
.8 

(2)  32.608000 
1.780125 

(4)_306487 
294315 

8.8 
.8 

26.72 
8.32 

34.388125 
1.808375 

W  36.196500 
.294610 

(5)- 121 72 
11038 

0.6 

.8 

(«)  35.0400 
.5625 

(«)-1134 
1104 

10.4 
.8 

35.6025 
.5650 

36.491110 
.295340 

(7)-30 
29 

(2)11.20 
5 

36.1675 
.5675 

W  36.7350 
912 

(*)  36.78645 
296 

(S)_l 

1 

11.25 
5 

36.78941 
296 

0 

11.30 
5 

36.8262 
912 

(5)  36.792 

(6)  36.79 

• 

11.35 
5 

36.9174 
912 

(7)  36.8 

(8)  4 

(3)11.40 
(*)1 

W  37.01 

(420,  Ex.  10) 


Horner's  method. 
The  second  root  is  found  as  follows : 

1 


279 


+  0. 
.6 

-12. 
+      .36 

-11.68 
+     .72 

-10.92 
+    1.08 

+  12 

-   6.984 

+  5.016 
-6.552 

-3  1.6060183069 
3.0096 

.6 
.6 

<!)  + 9600000000 
—  9569720304 

1.2 
.6 

a)- 1.536000000 
-       58953384 

(2) +  30279696 
—  16539179 

1.8 
6 

a) -9.840000 
-f       14436 

—  1.594953384 

-  58866552 

(3)4-13740517 
—  13232754 

(1)  2.400 
6 

-9.825564 
+       14472 

(2)  — 1.653819936 
-             97996 

W  + 507  763 
-496253 

2.406 
6 

-9.811092 
+       14508 

(2) -9.796584 
+             24 

-1.653917902 
-             97965 

(5)  +  11510 
—   9925 

2.412 
6 

(S)- 1.6540059 

-            784 

(«)  +  1585 
-1489 

2.418 
6 

—  9.796560 
+             24 

-1.6540943 

-             784 

+  96 

(«)  2.424 

—  9.796536 
24 

W-1.654173 
-               3 

<8)— 9.80 

• 
W-10 

-1.654176 
3 

(5)-1.65418 
<fi)     1.654 

(420,  Ex.  10) 


280 


NUMERICAL   EQUATIONS  OF   HIGHER   DEGREES. 


The  third  root  is  found  as  follows 


+  0 

-12. 

+  12 

-3  |.443276939ft 

A 

.16 

-  4.736 

2.9056 

A 

-11.84 

7.264             0) 

-.09440000 

A 

.32 

-4.308 

8864096 

.8 

-11.52 

(^)  2.656000 

(2)-57l9040000 

.4 

.48 

—  .438976 
2.217024 

5244710001 

1.2 

0)— 11.0400 

(8)_474329999 

.4 

656 
—  10.9744 

—  .436288 

342717760 

0)  1.60 

(2)  1.780736000 

W-131612239 

4 
1.64 

672 
—  10.9072 

—  32499333 
1.748236667 

119746687 
(5)-11865552 

4 

688 

—  32483439 

10259068 

1.68 

(2)- 10.838400 

(8)  1.715753228 

(6)-1606484 

4 

5289 

-2164430 

1538793 

1.72 

-10.833111 

1.713588798 

<7)-67691 

4 

5298 

—  2164360 

51293 

(«)  1.760 

-10.827813 

W  1.71142444 

(8)- 16398 

3 

5307 

-75749 

15388 

1.763 

W- 10.822506 

1.71076695 

W-IOIO 

3 

354 

-75748 

1026 

1.766 

-10.822152 

(5)  1.7099095 

3 

354 

-649 

1.769 

-10.821798 

1.7098446 

3 

354 

-649 

(3)  1.772 

W— 10.8214 

(«)  1.709780      • 

1 

-10 

(4)  2 

—  10.8213 

1.709770 

1 

-10 

-10.8212 
1 
W- 10.82 

(7)  1.60976 

(S)  1.7098 

(«)-ll 

(9)  1.710 

(430,  Ex.  10) 

HORNER'S   METHOD. 


281 


The  decimal  part  of  the  fourth  root  is  found  as  follows,  the  alter- 
nate signs  in  the  given  equation  being  changed : 


-0 

-12 

—  12 

—   3 

[3.9073785547 

3 

9 
—  3 

—   9 

-21               0 

-63 

3 

)-66.0000 

3 

18 
15 

45 

65.0241 

6 

(1)  24.000 

(2)_, 

.97590000 

3 

27 

48.249 

(3)_ 

.92562109 

9 

<^>  42.00 

72.249 

-5027891 

3 

11.61 

59.427 

3984354 

('>   12.0 

53.61 

(2)   131.676000 

(4). 

-1043537 

.9 

12.42 

.555584 

(-•5^ 

929889 

12.9 

66.03 

132.231584 

•-113648 

.9 

13.23 

.556349 

106278 

13.8 

<2)  79.2600 

<•''>   132.78793 

<6>-7370 

.9 

.1092 

2388 

6643 

14.7 

79.3692 

132.81181 

(7) -727 

.9 

.1092 

2388 

665 

<2>  15.60 

79.4784 

(*>   132.8357 

(S)-62 

.1092 

56 

53 

(8)    2 

(»>  79.59 

132.8413 

C9)— 9 

56 

9 

<*>  80. 

<-'^>   132.847 
<6>   132.85 
(7>   132.9 
(5)   133. 

0 

<5>   13.        —3. 

9073785547, root 

(420,  Ex.  10) 


282 


NUMERICAL   EQUATIONS   OF   HIGHER   DEGREES. 


11.  We  have  given, 

The  first  derived  polynomial  is 

X,=5x*  —  30x^-\-Q. 
Multiplying  X  by  5,  to  avoid  fractions, 


5x*— 50j:'4-30a;  +  5 
5a;*  — 30i:*+    Qx 


5x*  —  30x'-{-6 


.  —  20j;'  +  24a;  +  5. 

Hence, 
Multiplying  X,  by  4,  to  avoid  fractions, 


7?=20a:*  — 24ar— 5. 


20jr*  — 120j'  +  24 
20x*—   24.r'  — 5.C 


20j:»  — 24.r— 5 


—    96i:'  +  5j:  +  24 
llencc, 
Multiplying  i?  by  24, 

480x'—     5lex  —     120 
480x'—        2  ox'—      120.r 


:96x'  — oar— 24, 


96^:'- 5j:— 24 


5x,  +25 


25j:'—     456j:  —      120  ;  multiply  by  96 
2400j:'  — 43776^  -11520 
2400^'—      1252:  —      600 


i2,==43651a;  + 10920. 


—  43651jr— 10920 
Hence, 
Multiplying  i?,  by  43651, 

4190496x'—  218255.r—  1047624|43651.r  + 10920 

4190496ar*  + 


1048320jr 


'96.r,  —1266575 


—  1266575^:-  1047624;  multiply  by  43651 

—  55287265325a;-45729835224 

—  55287265325a;  — 13830999000 


_ 

-31898836224=— i?,. 

Therefore,  the  functions  are 

X  =x^~l0x'-\-6x—l, 

X,=52;*-302;'  +  6, 

R  =20x»-24i;-5, 

i2,=96x'-52;-24. 

iJj  =4365 1.C  + 10920, 

i23=31898836224. 

Let  a;=:  -  oo  ;  we  have      — 

+ 

—      +      —      + ,     5  variations. 

"   x=+  oo;       "              4- 

+ 

+      +      +      +,     0 

(420, 

Ex.  11) 

HORNER  S  METHOD. 


283 


Hence,  there  are  5  real  roots.  And  since  the  signs  of  the  given 
equation  present  two  variations,  two  of  the  roots  are  positive, 
{4L4LT  )  ;  consequently,  three  are  negative. 


To  ascertai 

n  the 

situation  of  the  roots,  let 

x=     0 

signs,     + 

+ 

— 

— 

+ 

+  , 

2 

variations. 

x=     1 

i( 

— 

— 

— 

+ 

+ 

+  , 

1 

variation. 

x=     2 

u 

— 

— 

+ 

4- 

+ 

+  , 

1 

u 

x=     3 

u 

— 

+ 

4- 

+ 

+ 

+  , 

1 

u 

x=z     4; 

u 

+ 

+ 

+ 

+ 

+ 

+  , 

0 

n 

«=-! 

u 

+ 

— 

— 

+ 

— 

+  , 

4 

variations. 

x=-2 

u 

+ 

— 

— 

+ 

— 

+, 

4 

a 

ar=-3 

u 

+ 

+ 

— 

+ 

■— 

+  , 

4 

u 

x=  —  4' 

(( 

— 

+ 

— 

+ 

— 

+  , 

5 

u 

Hence  we 

have 

1 

positive 

root  between 

0  . 

and 

1, 

1 

(( 

u 

u 

3  and 

4, 

2 

negative  roots 

a 

0  and  - 

-1, 

1        "        root       "        —3  and  -4. 

In  order  to  limit  still  further  the  roots  situated  between  0  and  1, 
and  0  and  —1,  we  may  employ  JT  alone,  according  to  ( 453,  3  ). 

As  x=zl  reduces  X  nearly  to  0,  the  positive  root  between  0  and  1 
must  be  nearly  1.  AVc  therefore  commence  with  x=ly  and  substi- 
tute the  descending  scale  of  tenths,  till  we  come  to  the  initial  figure. 
Thus, 

Substitute  x=z      1,  .9,  .8, 

The  signs  of  X  are  —  —  + 

Hence,  this  root  is  situated  between  .8  and  .9. 

For  the  roots  situated  between  0  and  —1,  we  proceed  thus: 
Substitute  x=  0,   —.1,   —.2,   —.3,   —.4,   —.5,   —.6,   —.7, 

The  signs  of  X  are         ++       —       —        —        —        —       + 

Hence,  one  root  is  between  —.1  and  —.2;  and  the  other  between 
—  .6  and  —.7. 

Thus,  we  find  the  initial  ficfures  of  the  several  roots,  taken  in  the 
order  of  their  algebraic  values,  to  be 

-3,      -.6,      -.1,      +.8,      4-3. 

Note. — In  extracting  the  negative  roots  in  this  example,  wo  may  change  the 
signs  of  the  alternate  terms,  remembering  always  to  supply  the  de&cieat  terms, 
with  0  for  coefficients. 

(420,  Ex.  11) 


284 


NUMERICAL   EQUATIONS   OF   HIGHER   DEGREES. 


The  operation  for  the  first  root  is 

as  follows  : 

1-0 

-10 

—  0 

+  6 

-  i;3.0653157913 

3 

9 
-1 

—  3 
-3 

-9 

-3                (^> 

-  9 

3 

-10.0000000000 

3 

18 

51 

144 

9.1254751776 

6 

17 

48 

141.00000000 

(2)-.8745248224 

3 

27 

132 

11.09125296 

.8222637421 

9 

44 

0)  180.000000 

152.09125296 

(3)- 522610803 

3 

36 

4.854216 

11.38577184 

496468108 

12 

0)  80.0000 

184.854216 

(2)163.47702480 

(*)-26142695 

3 

.9036 

4.908648 

.97572362 

16555014 

(1)  15.00 

80.9036 

189.762864 

164.45274842 

(5)-9587681 

6 

.9072 

4.963296 

.97781834 

8277654 

15.06 

81.8108 

(2)194.726160 

(3)165.4305668 

(6)- 1310027 

6 

.9108 

.418563 

588025 

1158879 

16.12 

82.7216 

195.144723 

165.4893093 

(7)-151148 

6 

.9144 

.418945 

588100 

148999 

15.18 

(2)83.6360 

195.563668 

(4)165.548179 

(S)-2149 

6 

765 

.419328 

1961 

1656 

15.24 

83.7125 

(8>195.9830 

165.550140 

(»)-493 

6 

765 

252 

1961 

497 

(2)  15.30 

83.7890 

196.0082 

(5)165.55210 

+  4 

765 

252 

98 

83.8655 

196.0334 

165.55308 

765 

252 

98 

C8>83.9 

(« 196.06 
(5)196. 

(6)2 

(6)165.5541 
1 
165.5542 
(7)165.554 
(8)165.55 

(»)165.6           —3.0653157913,  root 

(420, 

Ex.  11) 

HORNER'S  METHOD. 


285 


The  c 
1-0 
.6 

>peration  for 
-10 
.36 

-9.64 
.72 

-8.92 
1.08 

-7.84 
1.44 

the  second  root 

-  0 

-  5.784 

is  as  follows : 
+  6 
-3.4704 

— 1|.6915752805 
1.51776 

.6 
.6 

-  5.784 

—  5.352 

+  2.5296 
-6.6816 

ci)  +  . 51 77600000 
—  .5064468651 

1.2 
.6 

-11.136 
-   4.704 

(1)- 4.15200000 
-1.47518739 

(2)     113131349 
-71670641 

1.8 
.6 

(^)  — 15.840000 
—      .550971 

—  5.62718739 

—  1.52245656 

(3)     41460708 
—  35906031 

2.4 
.6 

(i)_  6.4000 
.2.781 

—  16.390971 

—  .525213 

<2)-7.14964395 
-       1742015 

(4)  5494677 
—  5042202 

0)  3.00 
9 

-6.1219 

.2862 

—  16.916184 

-  .498726 

-7.16706410 
—      1742538 

(5)  452475 
—  432268 

3.09 
9 

-5.8357 
2943 

(2)  — 17.414910 
—           5236 

(")  — 7.1844895 
-        87166 

(6)  20207 
-14409 

3.18 
9 

-5.5414 
.3024 

-17.420146 
-            5232 

-7.1932061 
-        87179 

C7)  5798 
-5764 

3.27 
9 

(2)-5.2390 
35 

-5.2355 
35 

-17.425378 
-           5229 

(4)-7.201924 
-         1221 

(S)  34 
-36 

3.36 
9 

(2)- 17.4306 
-           26 

—  7.203145 

-  1221 

-2 

(2)  3.45 

-6.2320 
35 

-17.4332 

-  26 

-  17.4358 

-  26 

(5)- 7.20437 
-           10 

-5.2285 
35 

-7.20447 
-          10 

—6.2 

(*>- 17.44 
(»)-l7. 

(«)-7.2046 
(7)-7.205 
(8)-7.21          -. 

,6915762805,  root. 

(420,  Ex.11) 


286  NUMERICAL   EQUATIONS   OF   HIGHER   DEGREES. 

The  operation  for  the  third  root  is  as  follows : 


1-  0 
.1 

.1 
.1 

.2 
.1 

.3 
.1 

.4 

.1 

0)    .50 

1 

.67 
1 

.64 

1 

.71 

7 

.78 

7 

(2)  .85 


10 


.01 


-9.99 
2 

—  9.97 

3 

—  9.94 

4 

(i)_9.9000 
399 

—  9.8601 

448 


-0 

—  .999 

—  .999 

—  .997 

—  1.996 

—  .994 

0)  — 2.990000 

—  .600207 


+  6. 

-  .0999 

6.9001 

—  .1996 


— 1.|.1756747993 
.59001 

0> -.4099900000 
.3810019857 


c^)     5.70050000    (2)  — 289880143 
—   .25761449  255583952 


5.44288551 
—   .30570946 


—  3.680207      (2)  5.13717605 

—  .687071        —       2549702 


(8) -34296191 
30496909 

(*)- 3799282 
3555530 


—  4.367278 

—  .683592 


5.11167903 
2573958 


<«  — 243752 
203158 


—  9.8153    (2)-5.050870      W  6.0859394 
497        —      48534        —         31213 


<6)_40594 
36562 


-9.7656   -5.099404     5.0828181 
546   —   48512   —    31248 


(7) -5042 
4571 


(2)-9.7ll0 


5.147916   (*>  5.079693 


43   —   48491 


363 


(8)-47l 
467 


9.7067  (8)_5.i964 


43 


58 


—  9.7024 
43 

—9.6981 
43 


5.2022 
58 

5.2080 

58 


5.079328 

—  365 

(•'^)  5.07896 

—  2 


(»)-14 
15 

+  1 


5.07894 
2 


(8) -9.7 


W-5.21 

(6)  5.0789 

(5)— 5. 

(7)  5.079 

(8)  6.08 

(9)  5.1 

(420, 

Ex.  11) 

-.1766747093,  root 


HORNER'S 

METHOD. 

2«7 

The  operation  for  the  fourth  root  is  as  follows : 

1  +0 
.8 

—  10. 
.64 

+  0 

-7.488 

+  6                   +1 1-8795087084 
-5.9904          -f   .00768 

.8 
.8 

-9.36 
1.28 

-7.488 
-6.464 

+      .0096          0)1.0076800000 
-11.1616            -.8742890793 

1.6 

.8 

—  8.08 
1.92 

-13.952 
—  4.928 

0)  — 11.15200000 
—   1.38784399 

(2).  1333909207 
-.1261650637 

2.4 
8 

-6.16 
2.56 

0)- 18.880000 
-     .232057 

—  12.48984399 

-  1.35266796 

(S)  72258570 
-71020747 

3.2 

.8 

0)- 3.6000 
.2849 

-19.112057 
-     .211771 

(2)  — 13.84251195 
—     .17582846 

W  1237823 
-1137128 

0)4.00 

7 

—  3.3151 

.2898 

-19.323828 
-      .191142 

-14.01834041 
—     .17601901 

(5)  100695 
-99500 

4.07 

7 

-3.0253 
.2947 

(2)-19.514970 
—         21525 

(8)— 14.1943594 

—  97899 

-14.2041493 

-  97905 

(6)  1195 
-1137 

4.14 

7 

-2.7306 
.2996 

-19.536495 
-         21173 

(7)  58 
-5^ 

4.21 

7 

(2)-2.4310 
392 

-19.557668 
-         20821 

W-14.213940 
-             157 

1 

4.28 

7 

-2.3918 
392 

(^)- 19.5  785 
-           12 

-14.214097 
-             157 

<2)  4.35 

-2.3526 
392 

-19.5797 
-           12 

(5)— 14.2143 

(6)-14.214 

(7)-14.2 

-2.3134 
392 

-19.5809 
—           12 

W-2.3 

(*)- 19.58 

(420,  Ex.11) 

288  NUMERICAL   EQUATIONS   OF   HIGHER   DEGREES. 

The  operation  for  the  fifth  root  is  as  follows ; 


1+0 
3 

3 

-10 
9 

—  1 

+  0 

—  3 

—  3 

+  6                             +1  |3.0535816265 

-9                        -9 

—  3                    0)- 8.0000000000 

3 

18 
17 

51 

48 

144 

7.5100940625 

6 

0)  141,00000000 

(2) -^.4899059375 

3 

27 
44 

132 

9.20188125 

.4805548729 

9 

0)180.000000 

150.20188125 

(3)- 93510646 

3 

36 

4.037625 

9.40565000 

80386391 

12 

0)  80.0000 

184.037025 

(2)159.60753125 

(4)  — 13124255 

3 

.7525 

4.075375 

.57742639 

12862718 

(1)15.00 

80.7525 

188.113000 

160.18495764 

(5)-261537 

5 

.7550 

4.113250 

.57817444 

160786 

15.05 

81.5075 

(2)  192.226250  C^)  160.7631321 

(G)_  100751 

5 

.7575 

.249212 

96489 

96472 

16.10 

82.2650 

192.475462 

160.7727810 

(^-4279 

5 

.7600 

.249350 

96491 

3216 

15.15 

(3)  83.0250 

192.724812  W  160.78243 

(8) -1063 

5 

458 

.249487 

154 

965 

15.20 

83.0708 

(8)192.9743 

160.78397 

(«)-98 

5 

458 

42 

154 

97 

(2)  15.25 

83.1166 

192.9785 

(^)  160.7855 

^1 

.458 

42 

(6)  160.786 
(7)160.79 

83.1624 

192.9827 

458 

42 

(8)160.8 

(3)  83.2 


(4)  193. 


(9)  161. 


(420,  Ex.  11) 


AN     \N1T1AU    FINE    °l^^^^o  return 

^^^  U   ASSESSED   !,°«    ''^Ce      THE  PENALTY 
TH^S    BOOK  ON  THE  OATE  °U  ^^^  ^„„„„ 

^«,LU.NCREASET0  50CEN^^^     ^^^^^^„     „^V 

DAY     AND     TO     S'"" 
OVERDUE. 


SFP     9  1933 


DEC  12    1938 


MAY  10  1940 


LD  21-^ 


;>     -ji 


